Random Matrices and K-theory for Exact C^{∗}-algebras

U. Haagerup^{∗}^{†} and S. Thorbjørnsen^{∗}

Abstract

In this paper we find asymptotic upper and lower bounds for the spectrum of random operators of the form

S^{∗}S = ( ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)})^{∗}( ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}),

where a_{1}, …, a_{r} are elements of an exact C^{∗}-algebra and Y_{1}^{(n)}, …, Y_{r}^{(n)} are complex Gaussian random n × n matrices, with independent entries. Our result can be considered as a generalization of results of Geman (1981) and Silverstein (1985) on the asymptotic behaviour of the largest and smallest eigenvalue of a random matrix of Wishart type. The result is used to give new proofs of:

(1) Every stably finite exact unital C^{∗}-algebra A has a tracial state.

(2) If A is an exact unital C^{∗}-algebra, then every state on K_{0}(A) is given by a tracial state on A.

The new proofs do not rely on quasitraces or on AW^{∗}-algebra techniques.

Introduction

Following the terminology in [HT], we let GRM(m, n, σ^{2}) denote the class of m×n random matrices B = (b_{ij})_{1≤i≤m, 1≤j≤n}, for which (Re(b_{ij}), Im(b_{ij}))_{1≤i≤m, 1≤j≤n} form a set of 2mn independent Gaussian random variables, all with mean 0 and variance \frac{1}{2}σ^{2}. In other words, the entries of B are mn independent complex random variables with distribution measure on ℂ given by

\frac{1}{πσ^{2}} exp(−\frac{|z|^{2}}{σ^{2}} ) dRe(z) dIm(z).

The theory of exact C^{∗}-algebras has been developed by Kirchberg (see [Ki1], [Ki2], [Ki3],

[Was] and references given there). A C^{∗}-algebra A is exact, if for all pairs (B, J), of a C^{∗}-algebra B and a closed two-sided ideal J in B, the sequence

0 ⟶ A ⊗_{min}J ⟶ A ⊗_{min}B ⟶ A ⊗_{min}(B/J ) ⟶ 0

∗Department of Mathematics and Computer Science, Odense University, Denmark. †MaPhySto - Centre for Mathematical Physics and Stochastics, funded by a grant from The Danish National Research Foundation.

is exact. Here, for any C^{∗}-algebras C and D, C ⊗_{min} D means the completion of the algebraic tensor product C ⊙ D in the minimal (=spatial) tensor norm. Sub-algebras and quotients of exact C^{∗}-algebras are again exact (cf. e.g. [Was, 2.5.2 and Corollary 9.3]), and the class of exact C^{∗}-algebras contains most of the C^{∗}-algebras of current interest, such as all nuclear C^{∗}-algebras, and the non-nuclear reduced group C^{∗}-algebras C_{r}^{∗}(𝔽_{n}), associated with the free group 𝔽_{n} on n generators (2 ≤ n ≤ ∞).

For any element T of a unital C^{∗}-algebra, we let sp(T) denote the spectrum of T. The main result of this paper is

0.1 Main Theorem. Let H and K be Hilbert spaces, and let a_{1}, …, a_{r} be elements of B(H, K), such that {a_{i}^{∗}a_{j} | 1 ≤ i, j ≤ r} is contained in an exact C^{∗}-subalgebra A of B(H). Let (Ω, F, P) be a fixed probability space, and let, for each n in ℕ, Y_{1}^{(n)}, …, Y_{r}^{(n)} be independent Gaussian random matrices on Ω in the class GRM(n, n, \frac{1}{n}). Put

S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}, (n ∈ ℕ),

and let c, d be positive real numbers. We then have

(i) If ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ ≤ c and ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ d, then for almost all ω in Ω,

lim sup_{n→∞} max [sp(S_{n}^{∗}(ω)S_{n}(ω))] ≤ (\sqrt{c} + \sqrt{d})^{2}.

(ii) If ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)}, ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ d, and d ≤ c, then for almost all ω in Ω,

lim inf_{n→∞} min [sp(S_{n}^{∗}(ω)S_{n}(ω))] ≥ (\sqrt{c} − \sqrt{d})^{2}. □

The Main Theorem can be considered as a generalization of the results of Geman (cf. [Gem]) and Silverstein (cf. [Si]), on the asymptotic behaviour of the largest and smallest eigenvalues of a random matrix of Wishart type (see also [BY], [YBK] and [HT]).

The Main Theorem has the following two immediate consquences:

0.2 Corollary. Let a_{1}, …, a_{r} be elements of an exact C^{∗}-algebra A, and for each n in ℕ, let Y_{1}^{(1)}, …, Y_{r}^{(n)} be independent elements of GRM(n, n, \frac{1}{n}). Then

lim sup_{n→∞} ∥ ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}(ω)∥ ≤ ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥^{\frac{1}{2}} + ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥^{\frac{1}{2}} ,

for almost all ω in Ω. □

0.3 Corollary. Let a_{1}, …, a_{r} and S_{n}, n ∈ ℕ, be as in the Main Theorem, and assume that ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ ≤ d, for some positive real numbers c, d, such that d < c. Then for almost all ω in Ω,

0 ∉ sp(S_{n}^{∗}(ω)S_{n}(ω)), eventually as n → ∞. □

In a subsequent paper [Th] by the second named author, it is proved, that if a_{1}, …, a_{r} and S_{n}, n ∈ ℕ, are as in the Main Theorem, and if furthermore ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and ∑_{i=1}^{r} a_{i}a_{i}^{∗} = d111_{B(K)}, for some positive real numbers c, d, then

lim_{n→∞} max [sp(S_{n}^{∗}S_{n})] = (\sqrt{c} + \sqrt{d})^{2}, almost surely,

and if c ≥ d, then

lim_{n→∞} min [sp(S_{n}^{∗}S_{n})] = (\sqrt{c} − \sqrt{d})^{2}, almost surely.

Hence the assymptotic upper and lower bounds in the Main Theorem cannot, in general, be improved.

Exactness is essential both for the Main Theorem and for the corollaries. An example of violation of the upper bound in the Main Theorem is given in Section 4. The example is based on the non-exact full C^{∗}-algebra C^{∗}(𝔽_{r}) associated with the free group on r generators, for r ≥ 6.

In [Haa], the first named author proved that bounded quasitraces on exact C^{∗}-algebras are traces. Together with results of Handelman (cf. [Han]) and Blackadar and Rørdam (cf. [BR]), this result implies

(1) Every stably-finite exact unital C^{∗}-algebra has a tracial state.

(2) If A is an exact unital C^{∗}-algebra, then every state on K_{0}(A) is given by a tracial state on A.

The proof in [Haa] of the above mentioned quasitrace result, relies heavily on ultra product techniques for AW^{∗}-algebras, but the starting point of the proof in [Haa] is the following fairly simple observation: Let a_{1}, …, a_{r} be r elements in a (not necessarily exact) C^{∗}- algebra A, such that ∑_{i=1}^{r} a_{i}^{∗}a_{i} = 111_{A} and ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ < 1. Let further x_{1}, …, x_{r} be a semi-circular system (in the sence of Voiculescu; cf. [Vo2]) in some C^{∗}-probability space (B, ψ). Then the operator s = ∑_{i=1}^{r} a_{i} ⊗ x_{i} in A ⊗ C^{∗}(x_{1}, …, x_{r},111_{B}), satisfies 0 ∉ sp(s^{∗}s) but 0 ∈ sp(ss^{∗}), and this implies that u = s(s^{∗}s)^{− \frac{1}{2}} is a non-unitary isometry in the C^{∗}-algebra A ⊗ C^{∗}(x_{1}, …, x_{r},111_{B}).

Corollary 0.3 can be viewed as a random matrix version of the result that 0 ∉ sp(s^{∗}s). The corresponding random matrix version of the result that 0 ∈ sp(ss^{∗}), holds too, i.e., if a_{1}, …, a_{r} and S_{n}, n ∈ ℕ, are as in Corollary 0.3, then with probability 1, 0 ∈ sp(S_{n}S_{n}^{∗}), eventually as n → ∞ (cf. [Th]). In view of Voiculescu’s random matrix model for a semicircular system (cf. [Vo1, Theorem 2.2]), it would have been more natural to substitute Y_{1}^{(n)}, …, Y_{r}^{(n)} from GRM(n, n, \frac{1}{n}), with a set of independent, selfadjoint Gaussian random matrices. However, we found it more tractable to work with the non-selfadjoint random matrices Y_{1}^{(n)}, …, Y_{r}^{(n)}.

In the last section (Section 9), we use Corollary 0.3 to give a new proof of the statements (1) and (2) above. The new proof does not rely on quasitraces or AW^{∗}-algebra techniques. The main step in the new proof of (1) and (2) is to prove, that Corollary 0.3 implies the following 0.4 Proposition. Let p, q be projections in an exact C^{∗}-algebra A, and assume that there exists an ϵ in ]0, 1[, such that

τ(q) ≤ (1 − ϵ)τ(p),

for all lower semi-continuous (possibly unbounded) traces τ: A_{+} → [0, ∞]. Then for some n in ℕ, there exists a partial isometry u in M_{n}(A) = A ⊗ M_{n}(ℂ), such that

u^{∗}u = q ⊗ 111_{M_{n}(ℂ)} and uu^{∗} ≤ p ⊗ 111_{M_{n}(ℂ)}. □

In the rest of this introduction, we shall briefly discuss the main steps of the proof of the Main Theorem. Observe first, that by a simple scaling argument, it is enough to treat the case d = 1. This normalization will be used throughout the paper. The proof of the Main Theorem relies on the following

0.5 Key Estimates. Let a_{1}, …, a_{r} be elements of B(H, K), let c be a positive constant, and put S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}, n ∈ ℕ, as in the Main Theorem. We then have

(a) If ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ ≤ c and ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ 1, then for 0 ≤ t ≤ min\{\frac{n}{2c}, \frac{n}{2}\},

𝔼[ exp(tS_{n}^{∗}S_{n})] ≤ exp ((\sqrt{c} + 1)^{2}t + (c + 1)^{2} \frac{t^{2}}{n} )111_{B(H^{n})}.  (0.1)

(b) If ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)}, ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)} and c ≥ 1, then for 0 ≤ t ≤ \frac{n}{2c},

𝔼[ exp(−tS_{n}^{∗}S_{n})] ≤ exp ( − (\sqrt{c} − 1)^{2}t + (c + 1)^{2} \frac{t^{2}}{n} )111_{B(H^{n})}.  □  (0.2)

We emphasize that the key estimates (0.1) and (0.2) hold without the exactness assumption of the Main Theorem. Once these estimates are proved, a fairly simple application of the Borel-Cantelli Lemma yields, that if H is finite dimensional, and λ_{max} and λ_{min} denote largest and smallest eigenvalues, then one has

lim sup_{n→∞} λ_{max}(S_{n}^{∗}S_{n}) ≤ (\sqrt{c} + 1)^{2}, almost surely,

in the situation of (a) above, and

lim inf_{n→∞} λ_{min}(S_{n}^{∗}S_{n}) ≥ (\sqrt{c} − 1)^{2}, almost surely,

in the situation of (b) above. (This is completely parallel to the proof of the complex version of the Geman-Silverstein result, given in [HT, Section 7]). To pass from the case dim(H) < ∞ to the case dim(H) = +∞, we need the assumption that the C^{∗}-algebra C^{∗}({a_{i}^{∗}a_{j} | 1 ≤ i, j ≤ r}) is exact, as well as the following characterization of exact C^{∗}-algebras, due to Kirchberg (cf. [Ki2] and [Was, Section 7]):

A unital C^{∗}-subalgebra A of B(H) is exact if and only if the inclusion map ι: A ↪ B(H) has an approximate factorization

A ⟶^{φ_{λ}} M_{n_{λ}}(ℂ) ⟶^{ψ_{λ}} B(H),

through a net of full matrix algebras M_{n_{λ}}(ℂ), λ ∈ Λ. Here, φ_{λ}, ψ_{λ} are unital completely positive maps, and

lim_{λ} ∥ψ_{λ} ∘ φ_{λ}(x) − x∥ = 0, for all x in A.

Finally, we use a dilation argument to pass from the condition ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{K} of (b) above, to the less restrictive one: ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ 1, which is assumed in (ii) of the Main Theorem (when d = 1). The proof of the fact that the key estimates (0.1) and (0.2) imply the Main Theorem, is given in Section 4 for the upper bound, and in Section 8 for the lower bound. Sections 1-3 and 5-7 are used to prove the key estimates (0.1) respectively (0.2).

In Section 1, we associate to any permutation π in the symmetric group S_{p}, a permutation π̂ in S_{2p}, for which π̂^{2} = π̂ ∘ π̂ = id and π̂(j) ≠ j for all j, namely the permutation given by

π̂(2j − 1) = 2π^{−1}(j),  (j ∈ {1, 2, …, p})
π̂(2j) = 2π(j) − 1,  (j ∈ {1, 2, …, p}).

Moreover, following [Vo1], we let ∼π̂ denote the equivalence relation on {1, 2, …, 2p}, generated by the expression:

j ∼π̂ π̂(j) + 1, (addition formed mod. 2p),

and we let d(π̂) denote the number of equivalence classes for ∼π̂. We can write d(π̂) = k(π̂) + l(π̂), where k(π̂) (resp. l(π̂)) denotes the number of equivalence classes for ∼π̂, consisting entirely of even numbers (resp. odd numbers) in {1, 2, …, 2p}. With this notation we prove, that for any random matrix B from GRM(m, n, 1),

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{π∈S_{p}} m^{k(π̂)}n^{l(π̂)}.  (0.3)

Consider next the quantity σ(π̂) = \frac{1}{2}(p+1−d(π̂)). It turns out, that σ(π̂) is always a nonnegative integer, and that σ(π̂) = 0 if and only if π̂ is non-crossing (cf. Definition 1.14). In Section 2 we show, that if a_{1}, …, a_{r} are elements of B(H, K) and S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}, where Y_{1}^{(n)}, …, Y_{r}^{(n)} are independent elements of GRM(n, n, \frac{1}{n}), then

𝔼[(S^{∗}S)^{p}] = ( ∑_{π∈S_{p}} n^{−2σ(π̂)} · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) ⊗ 111_{M_{n}(ℂ)}.  (0.4)

In [HT, Section 6], we found explicit formulas for the quantities 𝔼 ∘ Tr_{n}[exp(tB^{∗}B)] and 𝔼 ∘ Tr_{n}[B^{∗}B exp(tB^{∗}B)], where B is an element of GRM(m, n, 1). In Section 3, a careful comparison of the terms in (0.3) and (0.4), combined with these explicit formulas, allows us to prove, that if ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ ≤ c and ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ 1, then for 0 ≤ t ≤ min\{\frac{n}{2c}, \frac{n}{2}\},

∥𝔼[exp(tS^{∗}S)]∥ ≤ exp((c + 1)^{2} \frac{t^{2}}{n})∫_{0}^{∞} exp(tx) dμ_{c}(x),  (0.5)

where μ_{c} is the free (analog of the) Poisson distribution with parameter c (cf. [VDN] and [HT, Section 6]). The measure μ_{c} is also called the Marchenko-Pastur distribution (cf. [OP]), and it is given by

μ_{c} = max\{1 − c, 0\}δ_{0} + \frac{\sqrt{(x − a)(b − x)}}{2πx} · 1_{[a,b]}(x) dx,

where a = (\sqrt{c}−1)^{2}, b = (\sqrt{c}+1)^{2} and δ_{0} is the Dirac measure at 0. Since supp(μ_{c}) ⊆ [0, b], the first key estimate, (0.1), follows immediately from (0.5).

To prove the second key estimate, (0.2), we show in Sections 5-6, that under the condition

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)},  and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)},

one has, for any q in ℕ, the formula:

𝔼[P_{q}^{c}(S^{∗}S)] = [ ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}( ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}})] ⊗ 111_{M_{n}(ℂ)}.  (0.6)

Here P_{0}^{c}(x), P_{1}^{c}(x), P_{2}^{c}(x), …, is the sequence of monic polynomials obtained from 1, x, x^{2}, …, by the Gram-Schmidt orthogonalization process, w.r.t. the inner product

⟨f, g⟩ = ∫_{0}^{∞} f\overline{g} dμ_{c}, (f, g ∈ L^{2}(ℝ, μ_{c})).

Moreover, S_{q}^{irr} denotes the set of permutations ρ in S_{q}, for which

1 ≠ ρ(1) ≠ 2 ≠ ρ(2) ≠ ⋯ ≠ q ≠ ρ(q).

For fixed t in ℝ, we expand in Section 7 the exponential function x ↦ exp(tx), in terms of the polynomials P_{q}^{c}(x), q ∈ ℕ_{0}:

exp(tx) = ∑_{q=0}^{∞} ψ_{q}^{c}(t)P_{q}^{c}(x),  (x ∈ [0, ∞[).  (0.7)

We show that the coefficients ψ_{q}^{c}(t) are non-negative for all t in [0, ∞[, and that for any q in ℕ_{0},

|ψ_{q}^{c}(t)| ≤ (\frac{∫_{0}^{∞} exp(−tx) dμ_{c}(x)}{∫_{0}^{∞} exp(tx) dμ_{c}(x)}) · ψ_{q}^{c}(t),  (t ∈ [0, ∞[).  (0.8)

By combining (0.6), (0.7) and (0.8) with the proof of (0.5), we obtain that for c ≥ 1 and 0 ≤ t ≤ \frac{n}{2c},

∥𝔼[exp(−tS^{∗}S)]∥ ≤ exp((c + 1)^{2} \frac{t^{2}}{n})∫_{0}^{∞} exp(−tx) dμ_{c}(x),

and since supp(μ_{c}) ⊆ [a, ∞[ = [(\sqrt{c} − 1)^{2}, ∞[, when c ≥ 1, we obtain the second key estimate (0.2).

The rest of the paper is organized in the following way:

1. A Combinatorial Expression for 𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}], for a Gaussian Random Matrix B in GRM(m, n, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. A Combinatorial Expression for 𝔼[(S^{∗}S)^{p}] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

3. An Upper Bound for 𝔼[ exp(tS^{∗}S)], t ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4. Asymptotic Upper Bound on the Spectrum of S_{n}^{∗}S_{n} in the Exact Case . . . . . . . . .36

5. A New Combinatorial Expression for 𝔼[(S^{∗}S)^{p}] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

6. The Sequence of Orthogonal Polynomials for the Measure μ_{c} . . . . . . . . . . . . . . . . . . 60

7. An Upper Bound for 𝔼[ exp(−tS^{∗}S)], t ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

8. Asymptotic Lower Bound on the Spectrum of S_{n}^{∗}S_{n} in the Exact Case . . . . . . . . . 81

9. Comparison of Projections in Exact C^{∗}-algebras and States on the K_{0}-group . . 87

1 A Combinatorial Expression for 𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}], for a Gaussian Random Matrix B in GRM(m, n, 1)

For ξ in ℝ and σ^{2} in ]0, ∞[, we let N(ξ, σ^{2}) denote the Gaussian (or normal) distribution with mean ξ and variance σ^{2}. In [HT], we introduced the following class of Gaussian random matrices

1.1 Definition. (cf. [HT]) Let (Ω, F, P) be a classical probability space, let m, n be positive integers, and let

B = (b(i, j))\begin{matrix}1≤i≤m \\ 1≤j≤n \end{matrix}: Ω → M_{m,n}(ℂ),

be a complex, random m × n matrix defined on Ω. We say then that B is a (standard) Gaussian random m × n matrix with entries of variance σ^{2}, if the real valued random variables Re(b(i, j)), Im(b(i, j)), 1 ≤ i ≤ m, 1 ≤ j ≤ n, form a family of 2mn independent, identically distributed random variables, with distribution N(0, \frac{σ^{2}}{2} ). We denote by GRM(m, n, σ^{2}) the set of such random matrices defined on Ω. Note that σ^{2} equals the second absolute moment of the entries of an element from GRM(m, n, σ^{2}). □

In the following we shall omit mentioning the underlying probability space (Ω, F, P), and it will be understood that all considered random matrices/variables are defined on this probability space. As a matter of notation, by 1_{n} we denote the unit of M_{n}(ℂ), and by tr_{n} we denote the trace on M_{n}(ℂ) satisfying that tr_{n}(1_{n}) = 1. Moreover, we put Tr_{n} = n·tr_{n}.

Let B be an element of GRM(m, n, σ^{2}). Then for any p in ℕ, (B^{∗}B)^{p} is a positive definite n × n random matrix, and Tr_{n}((B^{∗}B)^{p}) is a positive valued, integrable, random variable. The main aim of this section is to derive a combinatorial expression for the moments 𝔼 ∘ Tr_{n}((B^{∗}B)^{p}) of B^{∗}B w.r.t. 𝔼 ∘ Tr_{n}, where 𝔼 denotes expectation w.r.t. P.

1.2 Lemma. Let m, n, r, p be positive integers, let B_{1}, B_{2}, …, B_{r} be independent elements of GRM(m, n, σ^{2}), and for each s in {1, 2, …, r}, let b(u, v, s), 1 ≤ u ≤ m, 1 ≤ v ≤ n, denote the entries of B_{s}. Then for any i_{1}, j_{1}, i_{2}, j_{2}, …, i_{p}, j_{p} in {1, 2, …, r}, we have that

\begin{matrix}𝔼 ∘ Tr_{n}(B_{i_{1}}^{∗}B_{j_{1}}B_{i_{2}}^{∗}B_{j_{2}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}) \\ = ∑_{\begin{matrix}1≤u_{2},u_{4},... ,u_{2p}≤m \\ 1≤u_{1},u_{3},... ,u_{2p−1}≤n \end{matrix}} 𝔼(\overline{b(u_{2}, u_{1}, i_{1})}b(u_{2}, u_{3}, j_{1}) ⋯ \overline{b(u_{2p}, u_{2p−1}, i_{p})}b(u_{2p}, u_{1}, j_{p})), \end{matrix} (1.1)

and moreover 𝔼 ∘ Tr_{n}(B_{i_{1}}^{∗}B_{j_{1}}B_{i_{2}}^{∗}B_{j_{2}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}) = 0, unless there exists a permutation π in the symmetric group S_{p}, such that j_{h} = i_{π(h)} for all h in {1, 2, …, p}.

Proof. Let f(u, v), 1 ≤ u ≤ m, 1 ≤ v ≤ n, denote the usual m × n matrix units, and let g(u, v), 1 ≤ u ≤ n, 1 ≤ v ≤ m, denote the usual n × m matrix units. We have then that

𝔼 ∘ Tr_{n}(B_{i_{1}}^{∗}B_{j_{1}}B_{i_{2}}^{∗}B_{j_{2}} ⋯ B_{i_{p}}^{∗}B_{j_{p}})

= ∑_{\begin{matrix}1≤v_{1},u_{2},v_{3},u_{4},... ,v_{2p−1},u_{2p}≤m \\ 1≤u_{1},v_{2},u_{3},v_{4},... ,u_{2p−1},v_{2p}≤n \end{matrix}} 𝔼(b^{∗}(u_{1}, v_{1}, i_{1})b(u_{2}, v_{2}, j_{1}) ⋯ b^{∗}(u_{2p−1}, v_{2p−1}, i_{p})b(u_{2p}, v_{2p}, j_{p}))

· Tr_{n}(g(u_{1}, v_{1})f(u_{2}, v_{2}) ⋯ g(u_{2p−1}, v_{2p−1})f(u_{2p}, v_{2p}))

= ∑_{\begin{matrix}1≤u_{2},u_{4},... ,u_{2p}≤m \\ 1≤u_{1},u_{3},... ,u_{2p−1}≤n \end{matrix}} 𝔼(\overline{b(u_{2}, u_{1}, i_{1})}b(u_{2}, u_{3}, j_{1}) ⋯ \overline{b(u_{2p}, u_{2p−1}, i_{p})}b(u_{2p}, u_{1}, j_{p})).

Note here, that for any u_{2}, u_{4}, …, u_{2p} in {1, 2, …, m} and u_{1}, u_{3}, …, u_{2p−1} in {1, 2, …, n}, we have because of the independence assumptions,

\begin{aligned}𝔼(\overline{b(u_{2}, u_{1}, i_{1})}b(u_{2}, u_{3}, j_{1}) & ⋯ \overline{b(u_{2p}, u_{2p−1}, i_{p})}b(u_{2p}, u_{1}, j_{p})) \\ & = ∏_{l=1}^{r} 𝔼( ∏_{h:i_{h}=l} \overline{b(u_{2h}, u_{2h−1}, l)} ∏_{h:j_{h}=l} b(u_{2h}, u_{2h+1}, l)), \end{aligned}

where 2h + 1 is calculated mod. 2p.

Note here, that for any l in {1, 2, …, r}, any u in {1, 2, …, m} and any v in {1, 2, …, n}, the distribution of b(u, v, l) is invariant under multiplication by complex numbers of norm 1. Hence, for any s, t in ℕ_{0}, 𝔼[\overline{b(u, v, l)^{s}} · b(u, v, l)^{t}] = 0, unless s = t. Using this, and the independence assumptions, it follows that for any l in {1, 2, …, r}, any u_{2}, u_{4}, …, u_{2p} in {1, 2, …, m} and any u_{1}, u_{3}, …, u_{2p−1} in {1, 2, …, n}, a necessary condition for the mean

𝔼( ∏_{h:i_{h}=l} \overline{b(u_{2h}, u_{2h−1}, l)} · ∏_{h:j_{h}=l} b(u_{2h}, u_{2h+1}, l))

to be distinct from zero is that

card({h ∈ {1, 2, …, p} | i_{h} = l}) = card({h ∈ {1, 2, …, p} | j_{h} = l}).  (1.2)

It follows that 𝔼∘Tr_{n}(B_{i_{1}}^{∗}B_{j_{1}}B_{i_{2}}^{∗}B_{j_{2}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}) = 0, unless (1.2) holds for all l in {1, 2, …, r}, and in this case, it is not hard to construct a permutation π from S_{p}, with the property described in the lemma. ■

1.3 Definition. Let p be a positive integer, and let π be an element of S_{p}. We associate to π a family Λ(π, m, n), m, n ∈ ℕ, of complex numbers, as follows:

Let B_{1}, B_{2}, …, B_{p} be independent elements of GRM(m, n, 1), and then define

Λ(π, m, n) = 𝔼 ∘ Tr_{n}(B_{1}^{∗}B_{π(1)}B_{2}^{∗}B_{π(2)} ⋯ B_{p}^{∗}B_{π(p)}). □

1.4 Remark. Let m, n, r, p be positive integers, and let B_{1}, B_{2}, …, B_{r} be arbitrary elements of GRM(m, n, σ^{2}). Moreover, let i_{1}, j_{1}, …, i_{p}, j_{p} be arbitrary elements of {1, 2, …, r}. We shall need the fact that the quantity 𝔼∘Tr_{n}(B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}) is bounded numerically by some constant K(m, n, p, σ^{2}) depending only on m, n, p, σ^{2} and not on r or the distributional relations between B_{1}, B_{2}, …, B_{r}. For this, adapt the notation from Lemma 1.2, and note then that by (1.1) from that lemma,

\begin{aligned}|𝔼 ∘ Tr_{n} & (B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}})| \\ & ≤ ∑_{\begin{matrix}1≤u_{2},u_{4},... ,u_{2p}≤m \\ 1≤u_{1},u_{3},... ,u_{2p−1}≤n \end{matrix}} | 𝔼(\overline{b(u_{2}, u_{1}, i_{1})}b(u_{2}, u_{3}, j_{1}) ⋯ \overline{b(u_{2p}, u_{2p−1}, i_{p})}b(u_{2p}, u_{1}, j_{p})) | . \end{aligned}

Then let M(2p, σ^{2}) denote the 2p’th absolute moment of the entries of an element from GRM(m, n, σ^{2}). A standard computation yields that M(2p, σ^{2}) = σ^{2p} · p!, but we shall not need this explicit formula. It follows now by the generalized Hölder inequality, that for any u_{2}, u_{4}, …, u_{2p} in {1, 2, …, m} and u_{1}, u_{3}, …, u_{2p−1} in {1, 2, …, n},

| 𝔼(\overline{b(u_{2}, u_{1}, i_{1})}b(u_{2}, u_{3}, j_{1}) ⋯ \overline{b(u_{2p}, u_{2p−1}, i_{p})}b(u_{2p}, u_{1}, j_{p})) |

≤ ∥\overline{b(u_{2}, u_{1}, i_{1})}∥_{2p}∥b(u_{2}, u_{3}, j_{1})∥_{2p} ⋯ ∥\overline{b(u_{2p}, u_{2p−1}, i_{p})}∥_{2p}∥b(u_{2p}, u_{1}, j_{p})∥_{2p}

= (M(2p, σ^{2})^{\frac{1}{2p}} )^{2p} = M(2p, σ^{2}).

Thus it follows that we may use K(m, n, p, σ^{2}) = m^{p}n^{p}M(2p, σ^{2}). □

1.5 Proposition. Let B be an element of GRM(m, n, 1), and let p be a positive integer. We then have

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{π∈S_{p}} Λ(π, m, n).

Proof. Let (B_{i})_{i∈ℕ} be a sequence of independent elements of GRM(m, n, 1). Note then that for any s in ℕ, the matrix \frac{1}{\sqrt{s}}(B_{1} + ⋯ + B_{s}) is again an element of GRM(m, n, 1), and therefore

\begin{aligned}𝔼 ∘ Tr_{n} [(B^{∗}B)^{p}] & = 𝔼 ∘ Tr_{n} [((s^{− \frac{1}{2}} (B_{1} + ⋯ + B_{s}))^{∗}(s^{− \frac{1}{2}} (B_{1} + ⋯ + B_{s})))^{p}] \\ & = s^{−p} ∑_{1≤i_{1},j_{1},... ,i_{p},j_{p}≤s} 𝔼 ∘ Tr_{n} [B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}] . \end{aligned} (1.3)

For π in S_{p} we define

M(π, s) = {(i_{1}, j_{1}, …, i_{p}, j_{p}) ∈ {1, 2, …, s}^{2p} | j_{1} = i_{π(1)}, …, j_{p} = i_{π(p)}}.

It follows then from Lemma 1.2, that in (1.3), we only have to sum over those 2p-tuples (i_{1}, j_{1}, …, i_{p}, j_{p}) that belong to M(π, s) for some π in S_{p}, and consequently

𝔼 ∘ Tr_{n} [(B^{∗}B)^{p}] = s^{−p} ∑_{(i_{1},j_{1},... ,i_{p},j_{p})∈∪_{π∈Sp}M(π,s)} 𝔼 ∘ Tr_{n} [B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}] .

Note though, that the sets M(π, s), π ∈ S_{p}, are not disjoint. However, if we put

D(s) = {(i_{1}, j_{1}, …, i_{p}, j_{p}) ∈ {1, 2, …, s}^{2p} | i_{1}, i_{2}, …, i_{p} are distinct},

then the sets M(π, s) ∩ D(s), π ∈ S_{p}, are disjoint. Thus we have

\begin{aligned}𝔼 ∘ Tr_{n} & [(B^{∗}B)^{p}] \\ & = s^{−p} ∑_{π∈S_{p}} ∑_{(i_{1},j_{1},... ,i_{p},j_{p})∈M(π,s)∩D(s)} 𝔼 ∘ Tr_{n} [B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}] \\ & + s^{−p} ∑_{(i_{1},j_{1},... ,i_{p},j_{p})∈(∪_{π∈Sp}M(π,s))\D(s)} 𝔼 ∘ Tr_{n} [B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}] . \end{aligned}  (1.4)

Note here, that if (i_{1}, j_{1}, …, i_{p}, j_{p}) ∈ M(π, s) ∩ D(s), then B_{i_{1}}, B_{i_{2}}, …, B_{i_{p}} are independent elements of GRM(m, n, 1), and hence

𝔼 ∘ Tr_{n} [B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}] = Λ(π, m, n).

Thus, the first term on the right hand side of (1.4) equals

s^{−p} ∑_{π∈S_{p}} card(M(π, s) ∩ D(s)) · Λ(π, m, n).

Here card(M(π, s) ∩ D(s)) = s(s − 1) ⋯ (s − p + 1), so

s^{−p} · card(M(π, s) ∩ D(s)) → 1 as s → ∞.

Hence, the first term on the right hand side of (1.4) tends to ∑_{π∈S_{p}} Λ(π, m, n) as s → ∞, and since the left hand side of (1.4) does not depend on s, it remains thus to show that the second term on the right hand side of (1.4) tends to 0 as s → ∞. This follows by noting that according to Remark 1.4, for any (i_{1}, j_{1}, …, i_{p}, j_{p}) in {1, 2, …, s}^{2p}, the quantity |𝔼 ∘ Tr_{n}[B_{i_{1}}^{∗}B_{j_{1}} ⋯ B_{i_{p}}^{∗}B_{j_{p}}]| is bounded by some constant K(m, n, p) depending only on m, n, p; not on s. And moreover,

s^{−p}card(( ∪_{π∈S_{p}} M(π, s)) \ D(s)) ≤ ∑_{π∈S_{p}} s^{−p}card(M(π, s) \ D(s))

= ∑_{π∈S_{p}} [s^{−p}card(M(π, s)) − s^{−p}card(M(π, s) ∩ D(s))]

= ∑_{π∈S_{p}} [1 − s^{−p}card(M(π, s) ∩ D(s))] → 0,

as s → ∞. This concludes the proof of the proposition. ■

It follows from Proposition 1.5, that in order to obtain a combinatorial expression for the moments 𝔼 ∘ Tr_{n}((B^{∗}B)^{p}) for a matrix B from GRM(m, n, 1), we need to derive a combinatorial expression for the quantities

Λ(π, m, n) = 𝔼 ∘ Tr_{n}(B_{1}^{∗}B_{π(1)}B_{2}^{∗}B_{π(2)} ⋯ B_{p}^{∗}B_{π(p)}),

where π ∈ S_{p} and B_{1}, …, B_{p} are independent elements of GRM(m, n, 1).

As it turns out, it shall be useful to have the relations between the factors in the product B_{1}^{∗}B_{π(1)}B_{2}^{∗}B_{π(2)} ⋯ B_{p}^{∗}B_{π(p)} determined in terms of a permutation π̂ in S_{2p}, rather than in terms of the permutation π from S_{p}.

1.6 Definition. Let p be a positive integer, and let π be a permutation in S_{p}. Then the permutation π̂ in S_{2p} is determined by the equations:

\begin{aligned}π̂(2i − 1) & = 2π^{−1}(i),  (i ∈ \{1, 2, …, p\}), \\ π̂(2i) & = 2π(j) − 1,  (i ∈ \{1, 2, …, p\}).  □ \end{aligned}

1.7 Remark. (a) Let p, π and π̂ be as in Definition 1.6. Note then that π̂^{2} = π̂ ∘ π̂ = id, the identity mapping on {1, 2, …, p}, and that π̂ maps odd numbers to even numbers, i.e., that π̂(j) − j = 1 (mod. 2), for all j in {1, 2, …, 2p}. In particular, π̂ has no fixed points. It is easy to check that {π̂ | π ∈ S_{p}} is exactly the set of permutations γ in S_{2p}, for which γ^{2} = id and γ(j) − 1 = 1 (mod. 2), for all j in {1, 2, …, 2p}. Moreover, the mapping π ↦ π̂ is injective.

(b) If B_{1}, B_{2}, …, B_{p} are independent elements of GRM(m, n, 1) , then we may write the product B_{1}^{∗}B_{π(1)}B_{2}^{∗}B_{π(2)} ⋯ B_{p}^{∗}B_{π(p)} in the form C_{1}^{∗}C_{2}C_{3}^{∗}C_{4} ⋯ C_{2p−1}^{∗}C_{2p}, where C_{2i−1} = B_{i} and C_{2i} = B_{π(i)} for all i in {1, 2, …, p}. Then π̂ is constructed exactly so that for any j, j′ in {1, 2, …, 2p}, we have

C_{j} = C_{j′} ⇔ j = j′ or π̂(j) = j′. □

1.8 Definition. We associate to π̂ an equivalence relation ∼π̂ on ℤ_{2p}. This is the equivalence relation (introduced by Voiculescu in [Vo1, Proof of Theorem 2.2]), generated by the expression:

j ∼π̂ π̂(j) + 1, (j ∈ {1, 2, …, 2p}),

where addition is formed mod. 2p. □

1.9 Remark. For a permutation π in S_{p}, the ∼π̂-equivalence classes are precisely the orbits in {1, 2, …, 2p} for the cyclic subgroup of S_{2p} generated by the permutation j ↦ π̂(j) + 1 (addition formed mod. 2p). Since this subgroup is finite, the equivalence class [j]π̂ of an element j in {1, 2, …, 2p} has the following form:

Let q be the number of elements in [j]π̂. Then

[j]π̂ = {j_{0}, j_{1}, …, j_{q−1}},

where j_{0} = j, j_{1} = π̂(j_{0}) + 1, j_{2} = π̂(j_{1}) + 1, …, j_{q−1} = π̂(j_{q−2}) + 1, j_{0} = π̂(j_{q−1}) + 1, (addition formed mod. 2p). □

It follows immediately from the definition of π̂ and Remark 1.9 that each ∼π̂-equivalence class consists entirely of even numbers or entirely of odd numbers. This is used in the following definition:

1.10 Definition. Let p be a positive integer, let π be a permutation in S_{p}, and consider the corresponding permutation π̂ in S_{2p}. By k(π̂) and l(π̂), we denote then the number of ∼π̂-equivalence classes consisting of even numbers, respectively the number of ∼π̂- equivalence classes consisting of odd numbers:

k(π̂) = card({[j]π̂ | j ∈ {2, 4, …, 2p}}),
l(π̂) = card({[j]π̂ | j ∈ {1, 3, …, 2p − 1}}).

Moreover, we define the quantities d(π̂) and σ(π̂) by the equations:

\begin{aligned}d(π̂) & = k(π̂) + l(π̂) = card(\{[j]π̂ | j ∈ \{1, 2, …, 2p\}\}), \\ σ(π̂) & = \frac{1}{2}(p + 1 − d(π̂)). □ \end{aligned}

Regarding the definition of σ(π̂), it will be shown later (cf. Theorem 1.13), that σ(π̂) is always a non-negative integer. The quantity d(π̂) was introduced by Voiculescu in [Vo1, Proof of Theorem 2.2].

1.11 Theorem. For any positive integers m, n and any π in S_{p}, we have that

Λ(π, m, n) = m^{k(π̂)}n^{l(π̂)}.

Proof. Consider independent elements B_{1}, B_{2}, …, B_{p} of GRM(m, n, 1), and for each j in {1, 2, …, p}, let b(u, v, j), 1 ≤ u ≤ m, 1 ≤ v ≤ n, denote the entries of B_{j}. It follows then by (1.1) in Lemma 1.2, that

\begin{matrix}Λ(π, m, n) \\ = 𝔼 ∘ Tr_{n}(B_{1}^{∗}B_{π(1)}B_{2}^{∗}B_{π(2)} ⋯ B_{p}^{∗}B_{π(p)}) \\ = ∑_{\begin{matrix}1≤u_{1},u_{3},...u_{2p−1}≤n \\ 1≤u_{2},u_{4},...u_{2p}≤m \end{matrix}} 𝔼(\overline{b(u_{2}, u_{1}, 1)}b(u_{2}, u_{3}, π(1)) ⋯ \overline{b(u_{2p}, u_{2p−1}, p)}b(u_{2p}, u_{1}, π(p))). \end{matrix} (1.5)

Due to the independence assumptions and [VDN, Lemma 4.1.3], it follows here that the term in the above sum corresponding to u_{1}, u_{2}, …, u_{2p} is zero, unless the corresponding matrix entries are pairwise conjugate to each other, i.e., unless we have that

b(u_{2i}, u_{2i+1}, π(i)) = b(u_{2π(i)}, u_{2π(i)−1}, π(i)),  (i ∈ {1, 2, …, p}).  (1.6)

Note also, that if (1.6) is satisfied, then the corresponding term in (1.5) equals 1, and consequently

Λ(π, m, n)
= card({(u_{1}, u_{2}, …, u_{2p}) | 1 ≤ u_{2i−1} ≤ n, 1 ≤ u_{2i} ≤ m, and (1.6) holds}). (1.7)

To calculate this cardinality, we note first that (1.6) is equivalent to the condition

u_{2i} = u_{2π(i)} and u_{2i+1} = u_{2π(i)−1},  (i ∈ {1, 2, …, p}),  (1.8)

where addition and subtraction are formed mod. 2p. Replacing now i by π^{−1}(i) in the first equation in (1.8), we get the equivalent condition:

u_{2i} = u_{2π^{−1}(i)} and u_{2i+1} = u_{2π(i)−1}, (i ∈ {1, 2, …, p}).

Recall then that by definition of π̂, π̂(2i − 1) = 2π^{−1}(i), and using this formula with i replaced by π(i), we get that also 2π(i) − 1 = π̂(π̂(2π(i) − 1)) = π̂(2i). Thus (1.6) is equivalent to the condition

u_{2i} = u_{π̂(2i−1)}, and u_{2i+1} = u_{π̂(2i)}, (i ∈ {1, 2, …, p}),

i.e., the condition

u_{j} = u_{π̂(j−1)}, (j ∈ {1, 2, …, 2p}).

Replacing finally j by π̂(j) + 1, we conclude that (1.6) is equivalent to the condition

u_{j} = u_{π̂(j)+1}, (j ∈ {1, 2, …, 2p}),

where π̂(j) + 1 is calculated mod. 2p. Having realized this, it follows immediately from Remark 1.9 and the definitions of k(π̂) and l(π̂), that the right hand side of (1.7) equals m^{k(π̂)}n^{l(π̂)}, and hence we have the desired formula. ■

1.12 Corollary. Let m, n be positive integers and let B be an element of GRM(m, n, 1). Then for any positive integer p, we have that

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{π∈S_{p}} m^{k(π̂)}n^{l(π̂)}.

Proof. This follows immediately by combining Proposition 1.5 and Theorem 1.11. ■

1.13 Theorem. Let p be a positive integer, and let π be a permutation in S_{p}. Then

(i) k(π̂) ≥ 1 and l(π̂) ≥ 1.

(ii) k(π̂) + l(π̂) ≤ p + 1.

(iii) σ(π̂) = \frac{1}{2}(p + 1 − k(π̂) − l(π̂)) is a non-negative integer.

Proof. (i) This is clear from Definition 1.10.

(ii) Since d(π̂) = k(π̂) + l(π̂) is the number of equivalence classes for ∼π̂, (ii) follows from [Vo1, Proof of Theorem 2.2].

(iii) The proof of (iii) requires more work. For elements p of ℕ and k, l of ℕ_{0}, we define

δ(p, k, l) = card({π ∈ S_{p} | k(π̂) = k and l(π̂) = l}).

By (i) and (ii), δ(p, k, l) = 0 unless k ≥ 1, l ≥ 1 and k + l ≤ p + 1. By Corollary 1.12, we have for an element B of GRM(m, n, 1), that

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)m^{k}n^{l}.

On the other hand, by the recursion formula for the moments 𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}], (p ∈ ℕ), found in [HT, Theorem 8.2], it follows that for p in ℕ, the moment 𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] can be expressed as a polynomial in m and n of the form:

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ′(p, k, l)m^{k}n^{l},

for suitable coeffecients δ′(p, k, l). By the remarks following the proof of [HT, Theorem 8.2], only terms of homogeneous degree p + 1 − 2j, j ∈ \{0, 1, 2, …, [\frac{p−1}{2}]\}, appear in this polynomial, i.e.,

δ′(p, k, l) = 0, when k + l = p (mod. 2).

If polynomials of two variables coincide on ℕ^{2}, then they are equal. Therefore, δ(p, k, l) = δ′(p, k, l) for all k, l, which proves that

card({π ∈ S_{p} | k(π̂) = k and l(π̂) = l}) = 0, if k + l = p (mod. 2).

Hence, σ(π̂) is an integer for all π in S_{p}, and by (ii), σ(π̂) ≥ 0. This proves (iii). ■

In the rest of this section, we shall introduce a method of “reductions of permutations”, which will be needed to determine the asymptotic lower bound of the spectrum of S_{n}^{∗}S_{n} (cf. Sections 5-8).

Let p be a positive integer, let π be a permutation in S_{p}, and consider the corresponding permutation π̂ in S_{2p}, introduced in Definition 1.6. Since π̂^{2} = id and π̂ has no fixed points, the orbits under the action of π̂ form a partition of {1, 2, …, 2p} into p sets, each with two elements.

1.14 Definition. Let p be a positive integer, and let π be a permutation in S_{p}. Following the standard definition of crossings in partitions of {1, 2, …, 2p} into sets of cardinality 2 (see e.g. [Sp]), we say that (a, b, c, d) is a crossing for π̂, if a, b, c, d ∈ {1, 2, …, 2p} such that

a < b < c < d,  and π̂(a) = c, π̂(b) = d.  (1.9)

If π̂ has no such crossings, we say that π̂ is a non-crossing permutation, and we let S_{p}^{nc} denote the set of permutations π in S_{p} for which π̂ is non-crossing. □

1.15 Definition. Let p be a positive integer, let π be a permutation in S_{p}, and let e be an element of {1, 2, …, 2p − 1}. We say then that (e, e + 1) is a pair of neighbours for π̂, if π̂(e) = e + 1. Note, that a pair of neighbours for π̂ is either of the form

(2k − 1, 2k), where k ∈ {1, …, p},

or of the form

(2k, 2k + 1), where k ∈ {1, …, p − 1}.

In the first case k = π(k), and in the second case π(k) = k + 1. □

1.16 Definition. Let p be a positive integer, let π be a permutation in S_{p}, and consider the permutation π̂ in S_{2p} introduced in Definition 1.6. We say then that π̂ is irreducible if π̂ has no pair of neighbours (in the sence of Definition 1.15), i.e., if π̂(j) ≠ j + 1 for all j in {1, 2, …, 2p − 1}. We denote by S_{p}^{irr} the set of permutations π in S_{p} for which π̂ is irreducible. Note that

π ∈ S_{p}^{irr} ⟺ 1 ≠ π(1) ≠ 2 ≠ π(2) ≠ ⋯ ≠ p ≠ π(p).

If π ∈ S_{p} \ S_{p}^{irr}, we say that π̂ is reducible. Note, that we do not require that π̂(2p) ≠ 1 in order for π̂ to be irreducible. Thus, irreducibility of π̂ is not invariant under cyclic permutations of {1, 2, …, 2p}. □

1.17 Lemma. Let p be a positive integer, and let π be a permutation in S_{p}^{nc}. Then π̂ has a pair of neighbours, i.e., π̂ is reducible in the sence of Definition 1.16. In other words, we have the inclusion S_{p}^{nc} ⊆ S_{p} \ S_{p}^{irr} or equivalently S_{p}^{irr} ⊆ S_{p} \ S_{p}^{nc}.

Proof. We prove the inclusion: S_{p}^{irr} ⊆ S_{p} \ S_{p}^{nc}. So let π from S_{p}^{irr} be given, and consider the set M = {j ∈ {1, 2, …, 2p} | π̂(j) ≥ j}. Note that M ≠ ∅, since clearly 1 ∈ M. Define now

α = min{π̂(j) − j | j ∈ M}.

Since π̂ has no fixed points and no pairs of neighbours (since π ∈ S_{p}^{irr}), we must have α ≥ 2. Choose j in {1, 2, …, 2p} such that π̂(j) − j = α. Since α ≥ 2, π̂(j) ≠ j + 1, or equivalently (since π̂^{2} = id), π̂(j + 1) ≠ j. Combining this with the definition of α, and the fact that π̂ has no fixed points, it follows that

π̂(j + 1) ∉ {j, j + 1, …, j + α} = {j, j + 1, …, π̂(j)},

i.e., either π̂(j + 1) < j or π̂(j + 1) > π̂(j). In the first case (π̂(j + 1), j, j + 1, π̂(j)) is a crossing for π̂, and in the second case (j, j + 1, π̂(j), π̂(j + 1)) is a crossing for π̂. In all cases, π ∈ S_{p} \ S_{p}^{nc}, as desired. ■

1.18 Definition. Let p be a positive integer, greater that or equal to 2, let π be a permutation in S_{p}, and assume that the permutation π̂ in S_{2p} has a pair (e, e + 1). Let φ be the order preserving bijection of {1, 2, …, 2p − 2} onto {1, 2, …, 2p} \ {e, e + 1}, i.e.,

φ(i) = \begin{cases}i, & if 1 ≤ i ≤ e − 1, \\ i + 2, & if e ≤ i ≤ 2p − 2. \end{cases}  (1.10)

By π_{0} we denote then the unique permutation in S_{p−1}, satisfying that

π̂_{0} = φ^{−1} ∘ π̂ ∘ φ.

We say that π̂_{0} is obtained from π̂ by cancellation of the pair (e, e + 1). □

A few words are appropriate about the introduction of π_{0} in the definition above. Note first of all that φ^{−1} ∘π̂ ∘φ is a well-defined permutation of {1, 2, …, 2p−2}, since π̂^{2} = id and π̂(e) = e + 1, so that π̂({1, 2, …, 2p} \ {e, e + 1}) = {1, 2, …, 2p} \ {e, e + 1}. To see that this permutation is actually of the form π̂_{0} for some (necessarily uniquely determined) permutation π_{0} in S_{p−1}, it suffices, by Remark 1.7(a), to check that (φ^{−1} ∘ π̂ ∘ φ)^{2} = id, and that φ^{−1}∘π̂∘φ(j)−j = 1 (mod. 2), for all j in {1, 2, …, 2p−2}. But these properties follow from the corresponding properties of π̂, and the fact that φ(j) = j (mod. 2), for all j.

1.19 Remark. Let p be a positive integer, greater that or equal to 2, let π be a permutation in S_{p}, and assume that the permutation π̂ in S_{2p} has a pair (e, e + 1). Let π_{0} be the permuation in S_{p−1} obtained from π as in Definition 1.18.

(a) If (e, e + 1) = (2k − 1, 2k) for some k in {1, …, p}, then π_{0} = ψ^{−1} ∘ π ∘ ψ, where ψ: {1, …, p − 1} → {1, …, p} \ {k} is the bijection given by

ψ(j) = \begin{cases}j, & if 1 ≤ j ≤ k − 1, \\ j + 1, & if k ≤ j ≤ p − 1. \end{cases}  (1.11)

(b) If (e, e + 1) = (2k, 2k + 1) for some k in {1, …, p − 1}, then π_{0} = χ^{−1} ∘ π ∘ ψ, where χ: {1, …, p − 1} → {1, …, p} \ {k + 1} is the bijection given by

χ(j) = \begin{cases}j, & if 1 ≤ j ≤ k, \\ j + 1, & if k + 1 ≤ j ≤ p − 1, \end{cases}  (1.12)

and where ψ is given by (1.11) □

1.20 Lemma. Let p be a positive integer, greater than or equal to 2, and let π be a permutation in S_{p} \ S_{p}^{irr}. Let (e, e + 1) be a pair for π̂ and let π_{0} be the permutation in S_{p−1}, for which π̂_{0} is the permutation obtained from π̂ by cancellation of (e, e + 1). Then π̂ is non-crossing if and only if π̂_{0} is non-crossing.

Proof. Let φ: {1, 2, …, 2p − 2} → {1, 2, …, 2p} \ {e, e + 1} be the bijection introduced in (1.10). We show that π̂_{0} has a crossing if and only if π̂ does.

Assume first that π̂_{0} has a crossing (a, b, c, d). Then since φ is (strictly) monotone and since (by definition of π_{0}) π̂(φ(a)) = φ(c), π̂(φ(b)) = φ(d), it follows that (φ(a), φ(b), φ(c), φ(d)) is a crossing for π̂.

Assume conversely that π̂ has a crossing (a′, b′, c′, d′). Then clearly

{a′, b′, c′, d′} ∩ {e, e + 1} = ∅,

so that the numbers φ^{−1}(a′), φ^{−1}(b′), φ^{−1}(c′), φ^{−1}(d′) are well-defined. It follows then, as above, that (φ^{−1}(a′), φ^{−1}(b′), φ^{−1}(c′), φ^{−1}(d′)) is a crossing for π̂_{0}. ■

1.21 Lemma. Let m, n be postive integers, and let B be an element of GRM(m, n, 1). Then

𝔼(B^{∗}B) = m111_{n},  and 𝔼(BB^{∗}) = n111_{m}.  (1.13)

Proof. Let (b_{ij})\begin{matrix}1≤i≤m \\ 1≤j≤n \end{matrix} be the entries of B. Then

𝔼(\overline{b_{ij}}b_{st}) = \begin{cases}1, & if (i, j) = (s, t), \\ 0, & otherwise. \end{cases}  (1.14)

Since (B^{∗}B)_{ij} = ∑_{s=1}^{m} \overline{b_{si}}b_{sj} and (BB^{∗})_{ij} = ∑_{s=1}^{m} b_{is}\overline{b_{js}}, for all i, j, (1.13) follows readily from (1.14). ■

1.22 Proposition. Let p be a positive integer, greater than or equal to 2, and let π be a permutation in S_{p} \ S_{p}^{irr}. Let (e, e + 1) be a pair for π̂ and let π_{0} be the permutation in S_{p−1}, for which π̂_{0} is the permutation obtained from π̂ by cancellation of (e, e + 1). Then with k(·), l(·), d(·) and σ(·) as introduced in Definition 1.10, we have that

(i) If e is odd, then k(π̂_{0}) = k(π̂) − 1 and l(π̂_{0}) = l(π̂).

(ii) If e is even, then k(π̂_{0}) = k(π̂) and l(π̂_{0}) = l(π̂) − 1.

In both cases, d(π̂_{0}) = d(π̂) − 1 and σ(π̂_{0}) = σ(π̂).

Proof. Let m, n be positive integers, and let B_{1}, …, B_{p} be independent random matrices from GRM(m, n, 1). By Theorem 1.11, we have then that

𝔼 ∘ Tr_{n}[B_{1}^{∗}B_{π(1)} ⋯ B_{p}^{∗}B_{π(p)}] = m^{k(π̂)}n^{l(π̂)}.  (1.15)

(i) Assume that e is odd, i.e., that (e, e + 1) = (2q − 1, 2q) for some q in {1, 2, …, p}. Then π(q) = q, and hence the set of random matrices

(B_{1}^{∗}, B_{π(1)}, …, B_{q−1}^{∗}, B_{π(q−1)}, B_{q+1}^{∗}, B_{π(q+1)}, …, B_{p}^{∗}, B_{π(p)})

is independent from the set (B_{q}^{∗}, B_{π(q)}). Therefore,

\begin{aligned}𝔼 ∘ Tr_{n} & [B_{1}^{∗}B_{π(1)} ⋯ B_{p}^{∗}B_{π(p)}] \\ & = 𝔼 ∘ Tr_{n}[B_{1}^{∗}B_{π(1)} ⋯ B_{q−1}^{∗}B_{π(q−1)}𝔼(B_{q}^{∗}B_{π(q)})B_{q+1}^{∗} ⋯ B_{p}^{∗}B_{π(p)}] \\ & = m · 𝔼 ∘ Tr_{n}[B_{1}^{∗}B_{π(1)} ⋯ B_{q−1}^{∗}B_{π(q−1)}B_{q+1}^{∗} ⋯ B_{p}^{∗}B_{π(p)}], \end{aligned}  (1.16)

where the last equality follows from Lemma 1.21. Note that only the random matrices B_{1}, …, B_{k−1}, B_{k+1}, …, B_{p} occur in the last expression in (1.16). Define now for i in {1, 2, …, p − 1},

B′_{i} = \begin{cases}B_{i}, & if 1 ≤ i ≤ k − 1, \\ B_{i+1}, & if k ≤ i ≤ p − 1. \end{cases}

Then by Remark 1.19(a), it follows that the last expression in (1.16) is equal to

m · 𝔼 ∘ Tr_{n}[(B′_{1})^{∗}B′_{π_{0}(1)} ⋯ (B′_{p−1})^{∗}B′_{π_{0}(p−1)}],

which, by Theorem 1.11, is equal to m · m^{k(π̂_{0})}n^{l(π̂_{0})}. Altogether, we have shown that

m^{k(π̂)}n^{l(π̂)} = m · m^{k(π̂_{0})}n^{l(π̂_{0})},

and since this holds for all positive integers m, n, it follows that k(π̂) = k(π̂_{0}) + 1 and l(π̂) = l(π̂_{0}). This proves (i).

(ii) Assume that e is even, i.e., that (e, e+1) = (2q, 2q+1), for some q in {1, 2, …, p−1}. Then π(q) = q + 1, and arguing now as in the proof of (i), we find that

\begin{aligned}m^{k(π̂)}n^{l(π̂)} & = 𝔼 ∘ Tr_{n}[B_{1}^{∗}B_{π(1)} ⋯ B_{p}^{∗}B_{π(p)}] \\ & = 𝔼 ∘ Tr_{n}[B_{1}^{∗}B_{π(1)} ⋯ B_{q}^{∗}𝔼(B_{π(q)}B_{q+1}^{∗})B_{π(q+1)} ⋯ B_{p}^{∗}B_{π(p)}] \\ & = n · 𝔼 ∘ Tr_{n}[B_{1}^{∗}B_{π(1)} ⋯ B_{q}^{∗}B_{π(q+1)} ⋯ B_{p}^{∗}B_{π(p)}], \end{aligned}  (1.17)

where the last equality follows from Lemma 1.21. Defining, this time, for each i in {1, 2, …, p − 1},

B′_{i} = \begin{cases}B_{i}, & if 1 ≤ i ≤ k, \\ B_{i+1}, & if k + 1 ≤ i ≤ p − 1, \end{cases}

we get by application of Remark 1.19(b), that the last expression in (1.17) is equal to

n · 𝔼 ∘ Tr_{n}[(B′_{1})^{∗}B′_{π_{0}(1)} ⋯ (B′_{p−1})^{∗}B′_{π_{0}(p−1)}],

which, by Theorem 1.11, equals n·m^{k(π̂)}n^{l(π̂)}. Arguing then as in the proof of (i), it follows that k(π̂) = k(π̂_{0}) and l(π̂) = l(π̂_{0}) + 1. This proves (ii).

The last statements of Proposition 1.22 follow immediately from (i), (ii) and Definition 1.10. ■

1.23 Proposition. Let p be a positive integer, and let π be a permutation in S_{p}. By finitely many (or possibly none) successive cancellations of pairs, π̂ can be reduced to either

(i) ê_{1}, where e_{1} is the trivial permutation in S_{1},

or

(ii) ρ̂, where ρ is a permutation in S_{q}^{irr} for some q in {2, …, p}.

Case (i) appears if and only if π ∈ S_{p}^{nc}.

Proof. It is clear, that by finitely many (or possibly none) successive cancellations of pairs, π̂ can be reduced to a permutation ρ̂, where either ρ ∈ S_{1} or ρ ∈ S_{q}^{irr} for some q in {2, 3, …, p}. By Lemma 1.20, π̂ is non-crossing if and only if ρ̂ is. Since S_{1} = S_{1}^{nc} = {e_{1}}, and S_{q}^{irr} ∩ S_{q}^{nc} = ∅ for all q in {2, 3, …, p}, by Lemma 1.17, it follows thus, that either case (i) or case (ii) occurs, and that case (i) occurs if and only if π̂ is non-crossing. ■

The following corollary is a special case of [Sh, Lemma 2.3]. For convenience of the reader, we include a proof based on Propositions 1.22 and 1.23.

1.24 Corollary. Let p be a positive integer and let π be a permutation in S_{p}. Then π̂ is non-crossing if and only if k(π̂) + l(π̂) = p + 1, or, equivalently, if and only if σ(π̂) = 0.

Proof. Assume first that π̂ is non-crossing. It follows then from Lemma 1.23, that by successive cancellations of pairs, π̂ may be reduced to ê_{1}, where e_{1} is the unique permutation in S_{1}. Since σ(·) is invariant under cancellations of pairs, (cf. Lemma 1.22), it follows that σ(π̂) = σ(ê_{1}), and it is straightforward to check that σ(ê_{1}) = 0.

Assume next that π̂ has a crossing. Then, by Lemma 1.23, there exist q in {2, …, p} and a permutation ρ in S_{q}^{irr}, such that π̂ may be reduced to ρ̂ by finitely many (or possibly none) successive cancellations of pairs. By Lemma 1.22, σ(π̂) = σ(ρ̂), and hence it suffices to show that σ(ρ̂) > 0, i.e., that d(ρ̂) < q + 1. Note for this, that since ρ̂ is irreducible, ρ̂(j) ≠ j + 1, for all j in {1, 2, …, 2q − 1}. Since ρ̂^{2} = id, this is equivalent to the condition that ρ̂(j) ≠ j − 1, for all j in {2, 3, …, 2q}, and by Remark 1.9, this implies that card([j]ρ̂) ≥ 2, for all j in {2, 3, …, 2q}. Letting r denote the number of ∼ρ̂-equivalence classes, that are distinct from [1]ρ̂, we have thus the inequality

2r + card([1]ρ̂) ≤ 2q.

Since r = d(ρ̂) − 1, and since card([1]ρ̂) ≥ 1, this implies that 2(d(ρ̂) − 1) + 1 ≤ 2q, and hence that d(ρ̂) ≤ q, as desired. ■

2 A Combinatorial Expression for the Moments of S^{∗}S

Let H and K be Hilbert spaces, let r be a positive integer, and let a_{1}, …, a_{r} be elements of B(H, K). Moreover, let n be a fixed positive integer, and let Y_{1}, …, Y_{r} be independent elements of GRM(n, n, \frac{1}{n}). We then define

S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}.

Note that S is a random variable taking values in B(H, K) ⊗ M_{n}(ℂ). The aim of this section is to derive combinatorical expressions for the moments

(id_{B(H)} ⊗ (𝔼 ∘ tr_{n}))[(S^{∗}S)^{p}] and (id_{B(H)} ⊗ 𝔼)[(S^{∗}S)^{p}], (p ∈ ℕ),

where id_{B(H)} denotes the identity mapping on B(H). Moreover, we shall obtain another combinatorical expression, which is an upper estimate for the norm of (id_{B(H)}⊗𝔼)[(S^{∗}S)^{p}]. For the sake of short notation, in the following we shall just write 𝔼 ∘ tr_{n} and 𝔼 instead of id_{B(H)} ⊗ (𝔼 ∘ tr_{n}) and id_{B(H)} ⊗ 𝔼.

We start with the following generalization of Proposition 1.5.

2.1 Proposition. Let H, K be Hilbert spaces, let r be a positive integer, and let a_{1}, …, a_{r} be elements of B(H, K). Moreover, let m, n be fixed positive integers, and let B_{1}, …, B_{r} be independent elements of GRM(m, n, 1). Then with T = ∑_{i=1}^{r} a_{i} ⊗ B_{i}, we have for any positive integer p, that

𝔼 ∘ Tr_{n}[(T^{∗}T)^{p}] = ∑_{π∈S_{p}} m^{k(π̂)}n^{l(π̂)} · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}.

Proof. Let (B(1, h))_{h∈ℕ}, …, (B(r, h))_{h∈ℕ} be sequences of elements from GRM(m, n, 1), such that (the entries of) the random matrices B(i, h), 1 ≤ i ≤ r, h ∈ ℕ, are jointly independent. Then for h in ℕ, we define

T_{h} = ∑_{i=1}^{r} a_{i} ⊗ B(i, h).

Note then, that for each s in ℕ,

s^{− \frac{1}{2}} ∑_{h=1}^{s} T_{h} = s^{− \frac{1}{2}} ∑_{h=1}^{s} ∑_{i=1}^{r} a_{i} ⊗ B(i, h) = ∑_{i=1}^{r} a_{i} ⊗ (s^{− \frac{1}{2}} ∑_{h=1}^{s} B(i, h)),

where the random matrices s^{− \frac{1}{2}} ∑_{h=1}^{s} B(1, h), …, s^{− \frac{1}{2}} ∑_{h=1}^{s} B(r, h) are independent elements of GRM(m, n, 1). It follows thus, that the moments of s^{−1}(∑_{h=1}^{s} T_{h})^{∗} ∑_{h=1}^{s} T_{h} w.r.t. 𝔼 ∘ Tr_{n} are equal to those of T^{∗}T. Thus for any p, s in ℕ, we have

\begin{aligned}𝔼 ∘ Tr_{n}[(T^{∗}T)^{p}] & = 𝔼 ∘ Tr_{n} [s^{−p}(( ∑_{h=1}^{s} T_{h})^{∗} ∑_{h=1}^{s} T_{h})^{p}] \\ & = s^{−p} · ∑_{\begin{matrix}1≤h_{1},h_{2},... ,h_{p}≤s \\ 1≤g_{1},g_{2},... ,g_{p}≤s \end{matrix}} 𝔼 ∘ Tr_{n} [T_{h_{1}}^{∗}T_{g_{1}}T_{h_{2}}^{∗}T_{g_{2}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}] . \end{aligned}  (2.1)

Consider here an arbitrary 2p–tuple (h_{1}, g_{1}, …, h_{p}, g_{p}) in {1, 2, …, s}^{2p}. Recalling then the definition of T_{h}, we find that

\begin{matrix}𝔼 ∘ Tr_{n} [T_{h_{1}}^{∗}T_{g_{1}}T_{h_{2}}^{∗}T_{g_{2}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}] \\ = ∑_{\begin{matrix}1≤i_{1},... ,i_{p}≤r \\ 1≤j_{1},... ,j_{p}≤r \end{matrix}}(a_{i_{1}}^{∗}a_{j_{1}} ⋯ a_{i_{p}}^{∗}a_{j_{p}}) · 𝔼[Tr_{n}(B(i_{1}, h_{1})^{∗}B(j_{1}, g_{1}) ⋯ B(i_{p}, h_{p})^{∗}B(j_{p}, g_{p}))]. \end{matrix}

Since B(i, h) is independent of B(j, g) unless i = j and h = g, it follows here from Lemma 1.2 in chapter 2, that

\begin{matrix}𝔼 ∘ Tr_{n} [B(i_{1}, h_{1})^{∗}B(j_{1}, g_{1}) ⋯ B(i_{p}, h_{p})^{∗}B(j_{p}, g_{p})] ≠ 0 \\ ⟹ ∃π ∈ S_{p} : (j_{1}, g_{1}) = (i_{π(1)}, h_{π(1)}), …, (j_{p}, g_{p}) = (i_{π(p)}, h_{π(p)}). \end{matrix}  (2.2)

In particular it follows that in (2.1), we only have to sum over (h_{1}, g_{1}, …, h_{p}, g_{p}) in ∪_{π∈S_{p}}M(π, s), where, as in the proof of Proposition 1.5 in chapter 2,

M(π, s) = {(h_{1}, g_{1}, …, h_{p}, g_{p}) ∈ {1, 2, …, s}^{2p} | g_{1} = h_{π(1)}, …, g_{p} = h_{π(p)}},

for any π in S_{p}. Following still the proof of Proposition 1.5 in chapter 2, we define,

D(s) = {(h_{1}, g_{1}, …, h_{p}, g_{p}) ∈ {1, 2, …, s}^{2p} | h_{1}, …, h_{p} are distinct },

and then the sets M(π, s) ∩ D(s), π ∈ S_{p}, are disjoint and

\begin{aligned}𝔼 ∘ Tr_{n} & [(T^{∗}T)^{p}] \\ & = s^{−p} ∑_{π∈S_{p}} ∑_{(h_{1},g_{1},... ,h_{p},g_{p})∈M(π,s)∩D(s)} 𝔼 ∘ Tr_{n}[T_{h_{1}}^{∗}T_{g_{1}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}] \\ & + s^{−p} ∑_{(h_{1},g_{1},... ,h_{p},g_{p})∈(∪_{π∈Sp}M(π,s))\D(s)} 𝔼 ∘ Tr_{n}[T_{h_{1}}^{∗}T_{g_{1}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}]. \end{aligned}  (2.3)

As was noted in the proof of Proposition 1.5, we have here that

s^{−p} · card(M(π, s) ∩ D(s)) → 1, as s → ∞, (π ∈ S_{p}),  (2.4)

and that

s^{−p} · card(( ∪_{π∈S_{p}} M(π, s)) \ D(s)) → 0, as s → ∞.  (2.5)

Moreover, for any h_{1}, g_{1}, …, h_{p}, g_{p} in ℕ, we have that

∥𝔼 ∘ Tr_{n}[T_{h_{1}}^{∗}T_{g_{1}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}]∥

≤ ∑_{\begin{matrix}1≤i_{1},... ,i_{p}≤r \\ 1≤j_{1},... ,j_{p}≤r \end{matrix}} ∥a_{i_{1}}^{∗}a_{j_{1}} ⋯ a_{i_{p}}^{∗}a_{j_{p}}∥ · |𝔼[Tr_{n}(B(i_{1}, h_{1})^{∗}B(j_{1}, g_{1}) ⋯ B(i_{p}, h_{p})^{∗}B(j_{p}, g_{p}))]|

≤ K(m, n, p, 1) · ∑_{\begin{matrix}1≤i_{1},... ,i_{p}≤r \\ 1≤j_{1},... ,j_{p}≤r \end{matrix}} ∥a_{i_{1}}^{∗}a_{j_{1}} ⋯ a_{i_{p}}^{∗}a_{j_{p}}∥,

where K(m, n, p, 1) is the constant introduced in Remark 1.4 in chapter 2. Since this constant does not depend on s, it follows thus, by (2.5), that the second term on the right hand side of (2.3) tends to 0 as s → ∞.

Regarding the first term on the right hand side of (2.3), for any π in S_{p} and any 2p–tuple (h_{1}, g_{1}, …, h_{p}, g_{p}) in M(π, s) ∩ D(s), we have that

\begin{matrix}𝔼 ∘ Tr_{n}[T_{h_{1}}^{∗}T_{g_{1}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}] = 𝔼 ∘ Tr_{n}[T_{h_{1}}^{∗}T_{h_{π(1)}} ⋯ T_{h_{p}}^{∗}T_{h_{π(p)}}] \\ = ∑_{\begin{matrix}1≤i_{1},... ,i_{p}≤r \\ 1≤j_{1},... ,j_{p}≤r \end{matrix}}(a_{i_{1}}^{∗}a_{j_{1}} ⋯ a_{i_{p}}^{∗}a_{j_{p}}) · 𝔼[Tr_{n}(B(i_{1}, h_{1})^{∗}B(j_{1}, h_{π(1)}) ⋯ B(i_{p}, h_{p})^{∗}B(j_{p}, h_{π(p)}))]. \end{matrix}

Recalling here the statement (2.2) and that h_{1}, …, h_{p} are distinct, it follows that the term in the above sum corresponding to (i_{1}, j_{1}, …, i_{p}, j_{p}) is 0, unless j_{1} = i_{π(1)}, …, j_{p} = i_{π(p)}. Thus we have that

𝔼 ∘ Tr_{n}[T_{h_{1}}^{∗}T_{g_{1}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}]

= ∑_{1≤i_{1},... ,i_{p}≤r}(a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}})

· 𝔼[Tr_{n}(B(i_{1}, h_{1})^{∗}B(i_{π(1)}, h_{π(1)}) ⋯ B(i_{p}, h_{p})^{∗}B(i_{π(p)}, h_{π(p)}))].

Note here, that since h_{1}, …, h_{p} are distinct, B(i_{1}, h_{1}), …, B(i_{p}, h_{p}) are independent for any choise of i_{1}, …, i_{p} in {1, 2, …, r}, and consequently

𝔼[Tr_{n}(B(i_{1}, h_{1})^{∗}B(i_{π(1)}, h_{π(1)}) ⋯ B(i_{p}, h_{p})^{∗}B(i_{π(p)}, h_{π(p)}))] = Λ(π, m, n),

for any i_{1}, …, i_{p} in {1, 2, …, r}. Thus, we may conclude that

𝔼 ∘ Tr_{n}[T_{h_{1}}^{∗}T_{g_{1}} ⋯ T_{h_{p}}^{∗}T_{g_{p}}] = Λ(π, m, n) · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}},

and this holds for any (h_{1}, g_{1}, …, h_{p}, g_{p}) in M(π, s) ∩ D(s). Therefore the first term on the right hand side of (2.3) equals

∑_{π∈S_{p}} s^{−p} · card(M(π, s) ∩ D(s)) · Λ(π, m, n) · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}},

and by (2.4), this tends to

∑_{π∈S_{p}} Λ(π, m, n) · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}},

as s → ∞. Since the left hand side of (2.3) does not depend on s, we get thus by letting s → ∞ in (2.3), that

𝔼 ∘ Tr_{n}[(T^{∗}T)^{p}] = ∑_{π∈S_{p}} Λ(π, m, n) · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}.

Combining finally with Theorem 1.11, we obtain the desired formula. ■

2.2 Corollary. Let H, K be Hilbert spaces, let r be a positive integer, and let a_{1}, …, a_{r} be elements of B(H, K). Moreover, let n be a fixed positive integer, and let Y_{1}, …, Y_{r} be independent elements of GRM(n, n, \frac{1}{n}). Then with S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}, we have for any positive integer p, that

𝔼 ∘ tr_{n}[(S^{∗}S)^{p}] = ∑_{π∈S_{p}} n^{−2σ(π̂)} · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}},  (2.6)

where σ(π̂) is the quantity introduced in Definition 2.3 in chapter 2.

Proof. With B_{i} = \sqrt{n} · Y_{i}, i ∈ \{1, 2, …, r\}, we have that B_{1}, …, B_{r} are independent elements of GRM(n, n, 1). It follows thus from Proposition 2.1, that for any p in ℕ,

n^{p} · 𝔼 ∘ Tr_{n}[(S^{∗}S)^{p}] = ∑_{π∈S_{p}} n^{k(π̂)+l(π̂)} ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}},

and consequently

𝔼 ∘ tr_{n}[(S^{∗}S)^{p}] = ∑_{π∈S_{p}} n^{−p−1+k(π̂)+l(π̂)} ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}.

Formula (2.6) now follows by noting that,

p + 1 − k(π̂) − l(π̂) = p + 1 − d(π̂) = 2σ(π̂),

for any π in S_{p}. ■

Our next objective is to derive a matrix version of formula (2.6). In other words, we shall obtain a combinatorial expression for 𝔼[(S^{∗}S)^{p}].

2.3 Lemma. Let n, r be positive integers and let Y_{1}, …, Y_{r} be independent elements of GRM(n, n, σ^{2}). Then for any (non–random) unitary n × n matrices u_{1}, …, u_{r}, the random matrices u_{1}Y_{1}u_{1}^{∗}, …, u_{r}Y_{r}u_{r}^{∗} are again independent elements of GRM(n, n, σ^{2}).

Proof. Note first that for each i in {1, 2, …, r}, the entries of u_{i}Y_{i}u_{i}^{∗} are all measurable w.r.t. the σ–algebra generated by the entries of Y_{i}. It follows therefore immediately that u_{1}Y_{1}u_{1}^{∗}, …, u_{r}Y_{r}u_{r}^{∗} are independent random matrices.

We note next, that it follows easily from Definition 1.1, that the joint distribution of the entries of an element from GRM(n, n, σ^{2}) has the following density w.r.t. Lebesgue measure on M_{n}(ℂ) ≃ ℝ^{2n^{2}}:

y ↦ (\frac{1}{πσ^{2}} )^{n^{2}} exp ( − \frac{1}{σ^{2}} · Tr_{n}(y^{∗}y)), (y ∈ M_{n}(ℂ)).  (2.7)

(Here the identification of M_{n}(ℂ) with ℝ^{2n^{2}} is given by y ↦ (Re(y_{jk}), Im(y_{jk}))_{1≤j,k≤n}). Now let u be a unitary n × n matrix, and consider then the linear mapping

Adu: y ↦ uyu^{∗}: M_{n}(ℂ) → M_{n}(ℂ).

Under the identification of M_{n}(ℂ) with ℝ^{2n^{2}}, the euclidian structure on ℝ^{2n^{2}} is given by the inner product:

⟨y, z⟩ = Re(Tr_{n}(z^{∗}y)), (y, z ∈ M_{n}(ℂ)).

Thus Adu: ℝ^{2n^{2}} → ℝ^{2n^{2}} is a linear isometry, and hence the Jacobi determinant of this mapping equals 1. Combining this fact with (2.7) and the usual transformation theorem for Lebesgue measure, it follows that for any Y in GRM(n, n, σ^{2}), the joint distribution of the entries of uY u^{∗} equals that of the entries of Y. ■

2.4 Lemma. Let B be a C^{∗}–algebra with unit 111, let n be a positive integer, and consider the tensor product B ⊗ M_{n}(ℂ). If x ∈ B ⊗ M_{n}(ℂ), such that (111 ⊗ u)x(111 ⊗ u)^{∗} = x for any unitary n × n matrix u, then x ∈ B ⊗ 111_{n}.

Proof. Assume that x ∈ B ⊗ M_{n}(ℂ), and that (111 ⊗ u)x(111 ⊗ u)^{∗} = x for any unitary n × n matrix u. Since M_{n}(ℂ) is the linear span of its unitaries, it follows that

x ∈ {y ∈ B ⊗ M_{n}(ℂ) | yT = T y for all T in 111 ⊗ M_{n}(ℂ)} = B ⊗ 111_{n},

where the last equality follows by standard matrix considerations; thinking of B ⊗ M_{n}(ℂ) as the set of n × n matrices with entries from B. ■

2.5 Proposition. Let S be as in Corollary 2.2. Then for any positive integer p, we have that:

𝔼[(S^{∗}S)^{p}] = ( ∑_{π∈S_{p}} n^{−2σ(π̂)} · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) ⊗ 111_{n}.

Proof. Let u be an arbitrary unitary n × n matrix, and define: S_{u} = ∑_{i=1}^{r} a_{i} ⊗ (uY_{i}u^{∗}). Note then that S_{u}S_{u}^{∗} = (111_{H} ⊗ u)S^{∗}S(111_{H} ⊗ u)^{∗}, where 111_{H} denotes the unit of B(H). It follows now by Lemma 2.3, that

𝔼[(S^{∗}S)^{p}] = 𝔼[(S_{u}^{∗}S_{u})^{p}] = 𝔼[(111_{H} ⊗ u)(S^{∗}S)^{p}(111_{H} ⊗ u)^{∗}] = (111_{H} ⊗ u)𝔼[(S^{∗}S)^{p}](111_{H} ⊗ u)^{∗}.

Since this holds for any unitary u, it follows from Lemma 2.4, that 𝔼[(S^{∗}S)^{p}] ∈ B(H)⊗111_{n}, and consequently

𝔼[(S^{∗}S)^{p}] = (tr_{n}(𝔼[(S^{∗}S)^{p}])) ⊗ 111_{n} = (𝔼 ∘ tr_{n}[(S^{∗}S)^{p}]) ⊗ 111_{n}.

The proposition now follows by application of Corollary 2.2. ■

In the next section, we shall obtain combinatorial expressions that are upper estimates for the moments 𝔼[(S^{∗}S)^{p}]. It follows from Proposition 2.5, that in order to obtain such combinatorial estimates, we should consentrate on deriving combinatorial estimates for the quantities

∥ ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} ∥,

where π ∈ S_{p}, and a_{1}, …, a_{r} are arbitrary bounded operators from a Hilbert space H to a Hilbert space K.

2.6 Definition. Let p be a positive integer, let π be a permutation in S_{p} and consider the permutation π̂ in S_{2p}. We then put

κ(π̂) = card({j ∈ {1, 3, …, 2p − 1} | π̂(j) > j}),
λ(π̂) = card({j ∈ {1, 3, …, 2p − 1} | π̂(j) < j}) + 1.  □

We note, that since π̂ has no fixed points, it follows that

κ(π̂) + λ(π̂) = p + 1, (p ∈ ℕ, π ∈ S_{p}).  (2.8)

Recalling that by definition of π̂, π̂(2h − 1) = 2π^{−1}(h) for all h in {1, 2, …, p}, it follows furthermore that

κ(π̂) = card({h ∈ {1, 2, …, p} | 2π^{−1}(h) > 2h − 1})
= card({h ∈ {1, 2, …, p} | π^{−1}(h) ≥ h})
= card({h ∈ {1, 2, …, p} | h ≥ π(h)}),

(2.9)

where the last equality follows by replacing h by π^{−1}(h). Similarly we have that

λ(π̂) = p + 1 − κ(π̂)
= card({h ∈ {1, 2, …, p} | π^{−1}(h) < h}) + 1
= card({h ∈ {1, 2, …, p} | h < π(h)}) + 1.

(2.10)

We note also, that since π̂(j) = j + 1 mod. 2 and π̂(π̂(j)) = j for all j, we have that

κ(π̂) = card(π̂[{j ∈ {1, 3, …, 2p − 1} | π̂(j) > j}])

= card({j ∈ {2, 4, …, 2p} | π̂(j) < j}), (2.11)

and similarly

λ(π̂) = card({j ∈ {2, 4, …, 2p} | π̂(j) > j}) + 1.  (2.12)

In connection with products of the form a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}, note that κ(π̂) denotes the number of h’s in {1, 2, …, p} for which the factor a_{i_{h}}^{∗} appears before the factor a_{i_{h}} in this product. Similarly λ(π̂) − 1 denotes the number of h’s in {1, 2, …, p} for which the factor a_{i_{h}} appears before the factor a_{i_{h}}^{∗}.

2.7 Proposition. Let H, K be Hilbert spaces, let r be a positive integer, and let a_{1}, …, a_{r} be elements of B(H, K). Let further c and d be positive real numbers, such that

∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ ≤ c and ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ d.  (2.13)

Then for any positive integer p and any permutation π in S_{p}, we have that

∥ ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} ∥ ≤ c^{κ(π̂)}d^{λ(π̂)−1}.

Proof. Let V be an infinite dimensional Hilbert space, and choose r isometries s_{1}, …, s_{r} in B(V), with orthogonal ranges, i.e.,

s_{i}^{∗}s_{j} = δ_{i,j}111_{B(V)},  (i, j ∈ {1, 2, …, r}).  (2.14)

Consider then the Hilbert space Ṽ = V ⊗⋯⊗V (p factors), and for each i in {1, 2, …, r} and h in {1, 2, …, p}, define the operator s(i, h) in B(Ṽ) by the equation

s(i, h) = 111_{B(V)} ⊗ ⋯ ⊗ 111_{B(V)} ⊗ S_{i} ⊗ 111_{B(V)} ⊗ ⋯ ⊗ 111_{B(V)}_{↑ h’th position}.  (2.15)

Next, put

t(i, h) = \begin{cases}s(i, h), & if h ≤ π^{−1}(h), \\ s(i, h)^{∗}, & if h > π^{−1}(h), \end{cases}  (i ∈ \{1, 2, …, r\}, h ∈ \{1, 2, …, p\}), (2.16)

and

A_{h} = ∑_{i=1}^{r} a_{i} ⊗ t(i, h),  (h ∈ {1, 2, …, p}).  (2.17)

We consider A_{h} as an element of B(H ⊗ Ṽ, K ⊗ Ṽ) in the usual way. We claim then that

A_{1}^{∗}A_{π(1)}A_{2}^{∗}A_{π(2)} ⋯ A_{p}^{∗}A_{π(p)} = ( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) ⊗ 111_{B(Ṽ)}.  (2.18)

To prove (2.18), observe first that

A_{1}^{∗}A_{π(1)} ⋯ A_{p}^{∗}A_{π(p)} = ∑_{\begin{matrix}1≤i_{1},i_{2},... ,i_{p}≤r \\ 1≤j_{1},j_{2},... ,j_{p}≤r \end{matrix}}(a_{i_{1}}^{∗}a_{j_{1}}a_{i_{2}}^{∗}a_{j_{2}} ⋯ a_{i_{p}}^{∗}a_{j_{p}}) ⊗ Π(i_{1}, j_{1}, i_{2}, j_{2}, …, i_{p}, j_{p}), (2.19)

where

Π(i_{1}, j_{1}, …, i_{p}, j_{p}) = t(i_{1}, 1)^{∗}t(j_{1}, π(1))t(i_{2}, 2)^{∗}t(j_{2}, π(2)) ⋯ t(i_{p}, p)^{∗}t(j_{p}, π(p)), (2.20)

for all i_{1}, j_{1}, …, i_{p}, j_{p} in {1, 2, …, r}. By (2.15) and (2.16), t(i, h) and t(i, h)^{∗} both commute with t(j, k) and t(j, k)^{∗}, as long as h ≠ k. Hence, we can reorder the factors in the product on the right hand side of (2.20), according to the second index in t(·, ·) and t(·, ·)^{∗}, in the following way

Π(i_{1}, j_{1}, …, i_{p}, j_{p}) = T (1)T (2) ⋯ T (p),

where

T (h) = \begin{cases}t(i_{h}, h)^{∗}t(j_{π^{−1}(h)}, h), & if h ≤ π^{−1}(h), \\ t(j_{π^{−1}(h)}, h)t(i_{h}, h)^{∗}, & if h > π^{−1}(h), \end{cases}

for each h in {1, 2, …, p}. By (2.16), it follows that

T (h) = \begin{cases}s(i_{h}, h)^{∗}s(j_{π^{−1}(h)}, h), & if h ≤ π^{−1}(h), \\ s(j_{π^{−1}(h)}, h)^{∗}s(i_{h}, h), & if h > π^{−1}(h), \end{cases}

and thus by (2.14)-(2.15), we get that for all i_{1}, j_{1}, …, i_{p}, j_{p} in {1, 2, …, r} and all h in {1, 2, …, p},

T (h) = \begin{cases}111_{B(Ṽ)}, & if i_{h} = j_{π^{−1}(h)}, \\ 0, & if i_{h} ≠ j_{π^{−1}(h)}. \end{cases}

Therefore, Π(i_{1}, j_{1}, …, i_{p}, j_{p}) = 0, unless i_{h} = j_{π^{−1}(h)}, for all h in {1, 2, …, p}, or equivalently, unless i_{π(h)} = j_{h}, for all h in {1, 2, …, p}, in which case Π(i_{1}, j_{1}, …, i_{p}, j_{p}) = 111_{B(Ṽ)}. Combining this with (2.19), we obtain (2.18).

Using again that s(i, h)^{∗}s(j, h) = δ_{i,j}111_{B(Ṽ)}, for all i, j in {1, 2, …, r}, we get that if h ≤ π^{−1}(h),

A_{h}^{∗}A_{h} = ∑_{i,j=1}^{r} a_{i}^{∗}a_{j} ⊗ s(i, h)^{∗}s(j, h) = ∑_{i=1}^{r} a_{i}^{∗}a_{i} ⊗ 111_{B(Ṽ)},

and if h > π^{−1}(h),

A_{h}A_{h}^{∗} = ∑_{i=1}^{r} a_{i}a_{i}^{∗} ⊗ 111_{B(Ṽ)}.

By (2.13), it follows thus, that

\begin{aligned}∥A_{h}∥^{2} & = ∥A_{h}^{∗}A_{h}∥ ≤ c,  if h ≤ π^{−1}(h), \\ ∥A_{h}∥^{2} & = ∥A_{h}A_{h}^{∗}∥ ≤ d,  if h > π^{−1}(h), \end{aligned}

so by (2.9) and (2.10),

∥A_{1}^{∗}A_{π(1)} ⋯ A_{p}^{∗}A_{π(p)}∥ ≤ ∏_{h=1}^{p} ∥A_{h}∥^{2} ≤ c^{κ(π̂)}d^{λ(π̂)−1}.

Together with (2.18), this proves the proposition. ■

2.8 Corollary. Let H, K be Hilbert spaces, let r be a positive integer, and let a_{1}, …, a_{r} be elements of B(H, K). Moreover, let n be a fixed positive integer, and let Y_{1}, …, Y_{r} be independent elements of GRM(n, n, \frac{1}{n}). Then with S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}, c = ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ and d = ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥, we have for any positiv integer p, that

∥𝔼[(S^{∗}S)^{p}]∥ ≤ ∑_{π∈S_{p}} n^{−2σ(π̂)}c^{κ(π̂)}d^{λ(π̂)−1}.

Proof. This follows immediately by combining Propositions 2.5 and 2.7. ■

In Section 3 we shall estimate further the quantity ∥𝔼[(S^{∗}S)^{p}]∥. As preparation for this, we will in Proposition 2.10 below, compare the numbers κ(π̂) and λ(π̂) with the numbers k(π̂) and l(π̂), defined in Section 1.

2.9 Lemma. Let p be a positive integer, let π be a permutation in S_{p}, and consider the permutation π̂ in S_{2p} and the corresponding equivalence relation ∼π̂. Then any equivalence class for ∼π̂, except possibly [1]π̂, contains an element j with the property that π̂(j) < j.

Proof. Let j′ be an element of {1, 2, …, 2p}, such that 1 ∉ [j′]π̂. We show that [j′]π̂ contains an element j such that π̂(j) < j. For this, note first, that we may assume that j′ is the smallest element of [j′]π̂. Then, by assumption, j′ ≥ 2. Now write in the usual manner

[j′]π̂ = {j_{0}, j_{1}, …, j_{q}}.

In particular, π̂(j_{q}) + 1 = j_{0} = j′ (addition formed mod. 2p). Now, since j′ ≥ 2, we have that j′ − 1 < j′, even when the subtraction is formed mod. 2p. Therefore, since j′ is the smallest element of [j′]π̂, π̂(j_{q}) = j′ − 1 < j′ ≤ j_{q}. Thus we may choose j = j_{q}. ■

2.10 Proposition. Let p be a positive integer, let π be a permutation in S_{p}, and consider the permutation π̂ in S_{2p}. We then have

(i) κ(π̂) ≥ k(π̂) and λ(π̂) ≥ l(π̂).

(ii) (κ(π̂) − k(π̂)) + (λ(π̂) − l(π̂)) = 2σ(π̂).

(iii) κ(π̂) = k(π̂) and λ(π̂) = l(π̂) if and only if π̂ is non–crossing.

Proof. (i) By Lemma 2.9 and the definition of l(π̂), it follows that

l(π̂) − 1 ≤ card({j ∈ {1, 3, …, 2p − 1} | π̂(j) < j}) = λ(π̂) − 1.

Similarly we find by application of (2.11), that

k(π̂) ≤ card({j ∈ {2, 4, …, 2p} | π̂(j) < j}) = κ(π̂).

(ii) We find by application of (2.8), that

(κ(π̂) − k(π̂)) + (λ(π̂) − l(π̂)) = (κ(π̂) + λ(π̂)) − d(π̂) = p + 1 − d(π̂) = 2σ(π̂).

(iii) This follows immediately by combining (i), (ii) and Corollary 1.24. ■

3 An upper bound for 𝔼[ exp(tS^{∗}S)], t ≥ 0

In the previous section, we computed 𝔼[(S^{∗}S)^{p}], for p in ℕ and S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}, where a_{1}, …, a_{r} ∈ B(H, K), for Hilbert spaces H and K, and where Y_{1}, …, Y_{r} are independent random matrices in GRM(n, n, \frac{1}{n}). For fixed p in ℕ, the function ω ↦ (S^{∗}(ω)S(ω))^{p} only takes values in a finite dimensional subspace of B(H, K) ⊗ M_{n}(ℂ). This is not the case for the function ω ↦ exp (tS^{∗}(ω)S(ω)), so in order to give precise meaning to the mean 𝔼[ exp(tS^{∗}S)], we will need the following definition (cf. [Ru, Definition 3.26]).

3.1 Definition. Let X be a Banach space, let (Ω, F, P) be a probability space, and let f: Ω → X be a mapping, that satisfies the following two conditions

(a) ∀φ ∈ X^{∗}: φ ∘ f ∈ L^{1}(Ω, F, P)

(b) ∃x_{0} ∈ X ∀φ ∈ X^{∗}: ∫_{Ω} φ ∘ f(ω) dP (ω) = φ(x_{0}).

We say then that f is integrable in X, and we call x_{0} the integral of f, and write

𝔼(f) = ∫_{Ω} f dP = x_{0}. □

Note, that in the above definition, x_{0} is uniquely determined by (b). Note also, that we do not require that ∫_{Ω} ∥f∥ dP < ∞, in order for f to be integrable. However, if X is finite dimensional, then this follows automatically from (a).

3.2 Proposition. Let H and K be Hilbert spaces, let a_{1}, …, a_{r} be elements of B(H, K), and let γ be a strictly positive number, such that

max {∥∑_{i=1}^{r}a_{i}^{∗}a_{i}∥, ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥} ≤ γ.

Furthermore, let n be a positive integer, let Y_{1}, …, Y_{r} be independent random matrices in GRM(n, n, \frac{1}{n}), and put S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}.

Then for any complex number t, such that |t| < \frac{n}{γ}, the function

ω ↦ exp (tS^{∗}(ω)S(ω)), (ω ∈ Ω),

is integrable in B(H^{n}), in the sence of Definition 3.1, and

𝔼[ exp(tS^{∗}S)] = ∑_{p=0}^{∞} \frac{t^{P}}{p!} 𝔼[(S^{∗}S)^{p}],  (3.1)

where the series on the right hand side is absolutely convergent in B(H^{n}).

Proof. By Proposition 2.5, we have for any p in ℕ,

𝔼[(S^{∗}S)^{p}] = ( ∑_{π∈S_{p}} n^{−2σ(π̂)} ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) ⊗ 111_{n},

and by Proposition 2.7 and formula (2.8), we have here for all π in S_{p}, that

∥ ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} ∥ ≤ γ^{p}.  (3.2)

Hence the abbsolute convergence of the right hand side of (3.1) will follow, if we can prove that

1 + ∑_{p=1}^{∞} \frac{(γ|t|)^{p}}{p!} (∑_{π∈S_{p}}n^{−2σ(π̂)}) < ∞,  (3.3)

whenever |t| < \frac{n}{γ}. For this, consider an element B of GRM(n, n, 1), and recall then from Corollary 1.12, that

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{π∈S_{p}} n^{k(π̂)+l(π̂)} = n^{p+1} ∑_{π∈S_{p}} n^{−2σ(π̂)}.

Hence for positive numbers s, we have

𝔼 ∘ Tr_{n}[ exp(sB^{∗}B)] = n(1 + ∑_{p=1}^{∞} \frac{(ns)^{p}}{p!} ∑_{π∈S_{p}}n^{−2σ(π̂)}).  (3.4)

From [HT, Theorem 6.4], we know that

𝔼 ∘ Tr_{n}[ exp(sB^{∗}B)] < ∞, when 0 ≤ s < 1.

Hence the sum in (3.4) is finite, whenever 0 ≤ s < 1, and this implies that (3.3) holds whenever |t| < \frac{n}{γ}.

Consider now the state space S(B(H)) of B(H) and an element φ of S(B(H)). For any ω in Ω, we have then that

φ[ exp (tS^{∗}(ω)S(ω))] = ∑_{p=0}^{∞} \frac{t^{p}}{p!} φ[(S^{∗}(ω)S(ω))^{p}],

which is clearly a positive measurable function of ω (since φ is a state). Moreover, by Lebesgues Monotone Convergence Theorem,

\begin{aligned}𝔼[φ( exp(tS^{∗}S))] & = ∑_{p=0}^{∞} \frac{t^{P}}{p!} 𝔼[φ((S^{∗}S)^{p})] \\ & = ∑_{p=0}^{∞} \frac{t^{P}}{p!} φ(𝔼[(S^{∗}S)^{p}]) \\ & = 1 + ∑_{p=0}^{∞} \frac{t^{p}}{p!} φ(∑_{π∈S_{p}}n^{−2σ(π̂)}( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) ⊗ 111_{n}) \\ & ≤ 1 + ∑_{p=0}^{∞} \frac{t^{p}}{p!} ∑_{π∈S_{p}}n^{−2σ(π̂)}∥ ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} ∥, \end{aligned} (3.5)

and by (3.2) and (3.3), the latter sum is finite, when |t| < \frac{n}{γ}. Since B(H)^{∗} = span(S(B(H)), it follows that the function ω ↦ exp(tS^{∗}(ω)S(ω)), is integrable, and (by the first two equalities in (3.5)) that 𝔼[ exp(tS^{∗}S)] is given by (3.1). ■

The main result of this section is the following

3.3 Theorem. Let H, K be Hilbert spaces, and let a_{1}, …, a_{r} be elements of B(H, K), satisfying that

∑_{i=1}^{r} a_{i}^{∗}a_{i} ≤ c111_{B(H)} and ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ 111_{B(K)},

for some constant c in ]0, ∞[. Consider furthermore independent elements Y_{1}, …, Y_{r} of GRM(n, n, \frac{1}{n}), and put S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}. Then for any t in [0, \frac{n}{2c}] ∩ [0, \frac{n}{2}], we have that

𝔼[ exp(tS^{∗}S)] ≤ exp ((\sqrt{c} + 1)^{2}t + (c + 1)^{2} · \frac{t^{2}}{n} ) · 111_{B(H^{n})}.

For the proof of Theorem 3.3, we need three lemmas. Before stating these lemmas, we introduce some notation:

For any p, k, l in ℕ, we put

δ(p, k, l) = card({π ∈ S_{p} | k(π̂) = k and l(π̂) = l}).  (3.6)

Note that for any p, k, l in ℕ, δ(p, k, l) = 0, unless k + l ≤ p + 1 (cf. Theorem 1.13).

For any complex number w and any n in ℕ_{0}, we put

(w)_{n} = \begin{cases}1, & if n = 0, \\ w(w + 1)(w + 2) ⋯ (w + n − 1), & if n ∈ ℕ. \end{cases}

We recall then, that the hyper-geometric function F, is defined by the formula

F (a, b, c; x) = ∑_{k=0}^{∞} \frac{(a)_{k}(b)_{k}}{(c)_{k}k!} x^{k},

for a, b, c, x in ℂ, such that c ∉ ℤ \ ℕ_{0}, and |x| < 1.

3.4 Lemma. For all positive real numbers α, β, we have that

∑_{p=1}^{∞} \frac{t^{p−1}}{(p − 1)!} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)α^{k−1}β^{l−1} = \frac{F (1 − α, 1 − β, 2, t^{2})}{(1 − t)^{α+β}}, (t ∈ ℂ, |t| < 1). (3.7)

Proof. Assume first that α = n and β = m, where m, n ∈ ℕ, and consider an element B of GRM(m, n, 1). Then by [HT, Theorem 6.4],

\frac{F (1 − n, 1 − m, 2, t^{2})}{(1 − t)^{m+n}} = \frac{1}{mn}𝔼 ∘ Tr_{n}[B^{∗}B exp(tB^{∗}B)] = \frac{1}{mn} ∑_{p=1}^{∞} \frac{t^{p−1}}{(p − 1)!}𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}].

But from Section 1 of this paper, we know that for any p in ℕ

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{π∈S_{p}} m^{k(π̂)}n^{l(π̂)} = ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)m^{k}n^{l},

and thus (3.7) holds for all α, β in ℕ. In particular, the left hand side (3.7) is finite for all α, β in ℕ. Since the left hand side of (3.7) is an increasing function of both α and β, it is therefore finite for all α, β in ]0, ∞[.

To prove (3.7) for general positive real numbers, α, β, we get first, as in [HT, Proof of Proposition 8.1], by multiplying the power series

F (1 − α, 1 − β, 2; t^{2}) = ∑_{j=0}^{∞} \frac{1}{j + 1}\binom{α − 1}{j}\binom{β − 1}{j}t^{2j}, (|t| < 1),

and

(1 − t)^{−(α+β)} = ∑_{k=0}^{∞} \binom{α + β + k − 1}{k}t^{k}, (|t| < 1),

that the power series expansion for \frac{F (1−α,1−β,2;t^{2})}{(1−t)^{α+β}} is given by

\frac{F (1 − α, 1 − β, 2; t^{2})}{(1 − t)^{α+β}} = ∑_{p=1}^{∞} ψ(p, α, β)t^{p−1},  (|t| < 1),  (3.8)

where for all p in ℕ,

ψ(p, α, β) = ∑_{j=0}^{[ \frac{p−1}{2} ]} \frac{1}{j + 1}\binom{α − 1}{j}\binom{β − 1}{j}\binom{α + β + p − 2j − 2}{p − 2j − 1}.  (3.9)

Since we know that (3.7) holds for all α, β in ℕ, we have, on the other hand, that

ψ(p, α, β) = \frac{1}{(p − 1)!} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)α^{k−1}β^{l−1},  (3.10)

for all α, β in ℕ. Thus, for fixed p, the right hand sides of (3.9) and (3.10) coincide whenever α, β ∈ ℕ, and since these two right hand sides are both polynomials in α and β, they must therefore coincide for all α, β in ]0, ∞[. In other words, (3.10) holds for all α, β in ]0, ∞[, and inserting this in (3.8), we get the desired formula. ■

3.5 Lemma. Let α, β be positive numbers, and assume that either α or β is an integer. Then

F (1 − α, 1 − β, 2; t^{2}) ≤ ∑_{j=0}^{∞} \frac{(αβ)^{j}t^{2j}}{j!(j + 1)!},  whenever 0 ≤ t < 1.  (3.11)

Proof. We recall first, that

F (1 − α, 1 − β, 2; t^{2}) = ∑_{j=0}^{∞} \frac{1}{j + 1}\binom{α − 1}{j}\binom{β − 1}{j}t^{2j}, (t ∈ [0, 1[).

If both α and β are integers, then

0 ≤ \binom{α − 1}{j} ≤ \frac{α^{j}}{j!} and 0 ≤ \binom{β − 1}{j} ≤ \frac{β^{j}}{j!} ,

for all j in ℕ_{0}, and (3.11) follows immediately. By symmetry of (3.11) in α and β, it is therefore sufficient to treat the case where α is an integer and β is not. In this case, we have

F (1 − α, 1 − β, 2; t^{2}) = ∑_{j=0}^{α−1} \frac{1}{j + 1}\binom{α − 1}{j}\binom{β − 1}{j}t^{2j}.

If β ≥ α, we have for any j in \{0, 1, …, α − 1\}, that 0 < \binom{α−1}{j} ≤ \frac{α^{j}}{j!} and 0 < \binom{β−1}{j} ≤ \frac{β^{j}}{j!} , and again (3.11) follows immediately.

Assume then that β < α, and let n be the integer for which n − 1 < β < n. Since α is an integer, and α > β, we have that α ≥ n. Forming now Taylor expansion on the function f(s) = F (1 − α, 1 − β, 2; s), (s > 0), it follows that

F (1 − α, 1 − β, 2; s) = ∑_{j=0}^{n−1} \frac{1}{j + 1}\binom{α − 1}{j}\binom{β − 1}{j}s^{j} + r_{n}(s),  (s > 0),  (3.12)

where r_{n}(s) = \frac{f^{(n)}(ξ(s))}{n!}s^{n}, for some ξ(s) in ]0, s[. It suffices thus to show that f^{(n)}(ξ) ≤ 0, for all ξ in [0, 1[, since this will imply that for all s in [0, 1[,

F (1 − α, 1 − β, 2; s) ≤ ∑_{j=0}^{n−1} \frac{1}{j + 1}\binom{α − 1}{j}\binom{β − 1}{j}s^{j},

where, as above, 0 < \binom{α−1}{j} ≤ \frac{α^{j}}{j!} and 0 < \binom{β−1}{j} ≤ \frac{β^{j}}{j!} , for all j in \{0, 1, …, n − 1\}.

To show that f^{(n)}(ξ) ≤ 0 for all ξ in [0, 1[, we note that by [HTF, Vol. 1, p. 58, formula (7)],

f^{(n)}(ξ) = \frac{d^{n}}{dξ^{n}} F (1 − α, 1 − β, 2; ξ) = \frac{(1 − α)_{n}(1 − β)_{n}}{(n + 1)!} F (n + 1 − α, n + 1 − β, n + 2; ξ),

for all ξ in [0, 1[. Note here that

(1 − α)_{n}(1 − β)_{n} = (α − 1)(α − 2) ⋯ (α − n)(β − 1)(β − 2) ⋯ (β − n) ≤ 0,

because α ≥ n and n − 1 < β < n. Moreover, by [HTF, Vol. 1, p. 105, formula (2)], we have for all ξ in [0, 1[

\begin{aligned}F (n + 1 − α, n + 1 − β, n + 2; ξ) & = (1 − ξ)^{α+β−n}F (α + 1, β + 1, n + 2; ξ) \\ & = (1 − ξ)^{α+β−n} ∑_{j=0}^{∞} \frac{(α + 1)_{j}(β + 1)_{j}}{j!(n + 2)_{j}} ξ^{j}, \end{aligned}

and therefore F (n + 1 − α, n + 1 − β, n + 2; ξ) > 0 for all ξ in [0, 1[. Taken together, it follows that f^{(n)}(ξ) ≤ 0 for all ξ in [0, 1[, as desired. ■

For any c in ]0, ∞[, we let μ_{c} denote the free Poisson distribution with parameter c, i.e., the probability measure on ℝ, given by

μ_{c} = max\{1 − c, 0\}δ_{0} + \frac{\sqrt{(x − a)(b − x)}}{2πx} · 1_{[a,b]}(x) · dx,  (3.13)

where a = (\sqrt{c} − 1)^{2}, b = (\sqrt{c} + 1)^{2} and δ_{0} is the Dirac measure at 0 (cf. [HT, Definition 6.5]).

3.6 Lemma. Let α, β be strictly positive real numbers, and assume that either α or β is an integer. Then for any t in [0, \frac{1}{2}],

1 + ∑_{p=1}^{∞} \frac{t^{p}}{p!} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)α^{k}β^{l−1} ≤ exp((α + β)t^{2}) ∫_{0}^{∞} exp(βtx) dμ\frac{α}{β} (x),

Proof. Using that − log(1 − t) = ∑_{n=1}^{∞} \frac{t^{n}}{n} ≤ t + t^{2}, whenever 0 ≤ t ≤ \frac{1}{2}, we note first that

(1 − t)^{−(α+β)} ≤ exp((α + β)t) exp((α + β)t^{2}), (t ∈ [0, \frac{1}{2}]).

Hence by Lemma 3.4 and Lemma 3.5,

∑_{p=1}^{∞} \frac{t^{p−1}}{(p − 1)!} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)α^{k−1}β^{l−1} ≤ exp((α + β)t) exp((α + β)t^{2}) ∑_{j=0}^{∞} \frac{(αβ)^{j}t^{2j}}{j!(j + 1)!} (3.14)

Put c = \frac{α}{β} and s = βt. From [HT, Formula (6.27)], it follows then that

∫_{0}^{∞} x exp(sx) dμ_{c}(x) = c exp((c + 1)s) ∑_{j=0}^{∞} \frac{c^{j}s^{2j}}{j!(j + 1)!} = c exp((α + β)t) ∑_{j=0}^{∞} \frac{(αβ)^{j}t^{2j}}{j!(j + 1)!}.

Hence (3.14) can be rewritten as

∑_{p=1}^{∞} \frac{t^{p−1}}{(p − 1)!} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)α^{k−1}β^{l−1} ≤ \frac{β}{α} exp((α + β)t^{2})∫_{0}^{∞}x exp(βtx) dμ\frac{α}{β} (x). (3.15)

Using then that \frac{t^{p}}{p!} = ∫_{0}^{t} \frac{u^{p−1}}{(p−1)!} du, for all p in ℕ, and that exp((α+β)u^{2}) ≤ exp((α+β)t^{2}), whenever 0 ≤ u ≤ t, we get by termwise integration of (3.15) (after replacing t by u), that

∑_{p=1}^{∞} \frac{t^{p}}{p!} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)α^{k−1}β^{l−1} ≤ \frac{β}{α} exp((α + β)t^{2})∫_{0}^{t}( ∫_{0}^{∞} x exp(βux) dμ\frac{α}{β} (x)) du

= \frac{β}{α} exp((α + β)t^{2})∫_{0}^{∞}x \frac{exp(βtx) − 1}{βx} dμ\frac{α}{β} (x)

= \frac{1}{α} exp((α + β)t^{2})∫_{0}^{∞}(exp(βtx) − 1) dμ\frac{α}{β} (x).

Hence, using that μ\frac{α}{β} is a probability measure, it follows that

1 + ∑_{p=1}^{∞} \frac{t^{p}}{p!} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)α^{k}β^{l−1} ≤ 1 + exp((α + β)t^{2})( ∫_{0}^{∞} exp(βtx) dμ\frac{α}{β} (x) − 1)

≤ exp((α + β)t^{2}) ∫_{0}^{∞} exp(βtx) dμ\frac{α}{β} (x).

This concludes the proof. ■

Proof of Theorem 3.3. Let a_{1}, …, a_{r}, Y_{1}, …, Y_{r} and S be as set out in Theorem 3.3. By Proposition 2.5 and Proposition 2.7, we have then that

\begin{aligned}𝔼[(S^{∗}S)^{p}] & = ( ∑_{π∈S_{p}} n^{−2σ(π̂)} ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) · 111_{n} \\ & ≤ ( ∑_{π∈S_{p}} n^{−2σ(π̂)}c^{κ(π̂)}) ⊗ 111_{B(H^{n})}, \end{aligned}  (3.16)

where κ(π̂) was introduced in Definition 2.6.

We assume first that c ≥ 1. By Proposition 2.10(i) and (ii), we have that

κ(π̂) ≤ k(π̂) + 2σ(π̂), (π ∈ S_{p}).

Hence,

𝔼[(S^{∗}S)^{p}] ≤ ( ∑_{π∈S_{p}} (\frac{n}{c})^{−2σ(π̂)}c^{k(π̂)}) · 111_{B(H^{n})}.

Using now that 2σ(π̂) = p + 1 − d(π̂) = p + 1 − k(π̂) − l(π̂), we find that

\begin{aligned}𝔼[(S^{∗}S)^{p}] & ≤ ((\frac{c}{n})^{p+1} ∑_{π∈S_{p}}n^{k(π̂)}(\frac{n}{c})^{l(π̂)}) · 111_{B(H^{n})} \\ & = ((\frac{c}{n})^{p} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}}δ(p, k, l)n^{k}(\frac{n}{c})^{l−1}) · 111_{B(H^{n})}, \end{aligned}

and therefore, for 0 ≤ t ≤ \frac{n}{max\{c,1\}} = \frac{n}{c}, it follows by application of Proposition 3.2, that

\begin{aligned}𝔼[ exp(tS^{∗}S)] & = 1 + ∑_{p=1}^{∞} \frac{t^{p}}{p!} 𝔼[(S^{∗}S)^{p}] \\ & ≤ (1 + ∑_{p=1}^{∞} \frac{1}{p!}(\frac{ct}{n})^{p} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}}δ(p, k, l)n^{k}(\frac{n}{c})^{l−1}) · 111_{B(H^{n})}. \end{aligned}

Using now Lemma 3.6, we get for 0 ≤ \frac{ct}{n} ≤ \frac{1}{2}, that

𝔼[ exp(tS^{∗}S)] ≤ ( exp ((n + \frac{n}{c})(\frac{ct}{n})^{2})∫_{0}^{∞} exp (\frac{n}{c}(\frac{ct}{n})x) dμ_{c}(x)) · 111_{B(H^{n})}
= ( exp (c(c + 1)\frac{t^{2}}{n} )∫_{0}^{∞} exp(tx) dμ_{c}(x)) · 111_{B(H^{n})}
≤ ( exp ((c + 1)^{2} · \frac{t^{2}}{n} )∫_{0}^{∞} exp(tx) dμ_{c}(x)) · 111_{B(H^{n})}.

Since supp(μ_{c}) ⊆ [0, (\sqrt{c} + 1)^{2}], it follows that

𝔼[ exp(tS^{∗}S)] ≤ exp ((c + 1)^{2} · \frac{t^{2}}{n} ) exp ((\sqrt{c} + 1)^{2}t) · 111_{B(H^{n})},

and this proves the theorem in the case where c ≥ 1. Assume then that c < 1. In this case we use (3.16) together with the fact that κ(π̂) ≥ k(π̂) for all π in S_{p}, (Proposition 2.10(ii)) to obtain

𝔼[(S^{∗}S)^{p}] ≤ ( ∑_{π∈S_{p}} n^{−2σ(π̂)}c^{k(π̂)}) · 111_{B(H^{n})}

≤ (\frac{1}{n^{p+1}} ∑_{π∈S_{p}}(nc)^{k(π̂)}n^{l(π̂)}) · 111_{B(H^{n})}

= (\frac{1}{n^{p}} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}}δ(p, k, l)(nc)^{k}n^{l−1}) · 111_{B(H^{n})}.

Hence for 0 ≤ t < \frac{n}{max\{c,1\}} = n, we get by application of Proposition 3.2,

\begin{aligned}𝔼[ exp(tS^{∗}S)] & ≤ 1 + ∑_{p=1}^{∞} \frac{t^{p}}{p!}𝔼[(S^{∗}S)^{p}] \\ & ≤ (1 + ∑_{p=1}^{∞} \frac{1}{p!}(\frac{t}{n})^{p} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}} δ(p, k, l)(nc)^{k}n^{l−1}) · 111_{B(H^{n})}. \end{aligned}

Hence by Lemma 3.6, we have for 0 ≤ \frac{t}{n} ≤ \frac{1}{2},

𝔼[ exp(tS^{∗}S)] ≤ ( exp ((nc + n)(\frac{t}{n})^{2})∫_{0}^{∞} exp (n(\frac{t}{n})x) dμ_{c}(x)) · 111_{B(H^{n})}
= ( exp ((c + 1)\frac{t^{2}}{n} )∫_{0}^{∞} exp(tx) dμ_{c}(x)) · 111_{B(H^{n})}
≤ exp ((c + 1)^{2} · \frac{t^{2}}{n})) exp ((\sqrt{c} + 1)^{2}t) · 111_{B(H^{n})},

and this completes the proof. ■

3.7 Remark. Assume that a_{1}, …, a_{r} ∈ B(H, K), satisfying that ∑_{i=1}^{r} a_{i}^{∗}a_{i} ≤ c111_{B(H)} and ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ d111_{B(H)}, for some positive constants c and d. Consider furthermore independent elements Y_{1}, …, Y_{r} of GRM(n, n, \frac{1}{n}), and put S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}. Applying then Theorem 3.3 to a′_{i} = \frac{1}{\sqrt{d}}a_{i} and c′ = \frac{c}{d}, we get the following extension of Theorem 3.3:

For any t in [0, \frac{n}{2c}] ∩ [0, \frac{n}{2d}],

𝔼[ exp(tS^{∗}S)] ≤ exp ((\sqrt{c} + \sqrt{d})^{2}t + (c + d)^{2} · \frac{t^{2}}{n} ) · 111_{B(H^{n})}. □

4 Asymptotic Upper Bound on the Spectrum of S_{n}^{∗}S_{n} in the Exact Case

Throughout this section, we consider elements a_{1}, …, a_{r} of B(H, K) (for Hilbert spaces H and K), satisfying that

∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ ≤ c, and ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ 1,  (4.1)

for some constant c in ]0, ∞[. Let A denote the unital C^{∗}-subalgebra of B(H) generated by the family {a_{i}^{∗}a_{j} | i, j ∈ {1, …, r}} ∪ {111_{B(H)}}. Furthermore, for each n in ℕ, we consider independent elements Y_{1}^{(n)}, …, Y_{r}^{(n)} of GRM(n, n, \frac{1}{n}), and we define

S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}.  (4.2)

In this section, we shall determine (almost surely) the asymptotic behaviour (as n → ∞) of the largest element of the spectrum of S_{n}^{∗}S_{n} (i.e., the norm of S_{n}^{∗}S_{n}), under the assumption that A is an exact C^{∗}-algebra. We start by studying the corresponding asymptotic behaviour for the image of S_{n}^{∗}S_{n} under certain matrix valued completely positive mappings. More precisely, let d be a fixed positive integer, and let Φ: A → M_{d}(ℂ) be a unital completely positive mapping. For each n in ℕ, let id_{n}: M_{n}(ℂ) → M_{n}(ℂ) denote the identity mapping on M_{n}(ℂ). We then define

V_{n} = (Φ ⊗ id_{n})(S_{n}^{∗}S_{n}) = ∑_{i,j=1}^{r} Φ(a_{i}^{∗}a_{j}) ⊗ (Y_{i}^{(n)})^{∗}Y_{j}^{(n)},  (n ∈ ℕ).  (4.3)

Note that V_{n} is a random variable taking values in M_{d}(ℂ) ⊗ M_{n}(ℂ) ≃ M_{dn}(ℂ). As indicated above, our first objective is to determine the asymptotic behaviour of the largest eigenvalue of V_{n}. We emphasize, that this step does not require that A be exact.

The following lemma is a version of Jensen’s Inequality, which we shall need significantly in this section and in Section 8. The lemma has been proved in much more general settings by Brown and Kosaki (cf. [BK]) and by Petz (cf. [Pe]). For the readers convenience, we include a short proof, handling the special case needed here.

4.1 Lemma. (i) Let L be a Hilbert space, and let P be a finite dimensional projection in B(L). Let tr denote the normalized trace on B(P (L)). Then for any selfadjoint element a of B(L), and any convex function g: ℝ → ℝ, we have that

tr[g(P aP)] ≤ tr[P g(a)P ].  (4.4)

(ii) Let B be a C^{∗}-algebra, let m be a positive integer and let Ψ: B → M_{m}(ℂ) be a unital completely positive mapping. Then for any selfadjoint element a of B and any convex function g: ℝ → ℝ, we have that

tr_{m}[g(Ψ(a))] ≤ tr_{m}[Ψ(g(a))].

Proof. (i) Note first that g is continuous (being convex on the whole real line). Let m denote the dimension of P (L), and choose an orthonormal basis (e_{1}, …, e_{m}) for P (L) consisting of eigenvectors for P aP. Let λ_{1}, …, λ_{m} be the corresponding eigenvalues for P aP, i.e.,

λ_{i} = ⟨P aP e_{i}, e_{i}⟩ = ⟨ae_{i}, e_{i}⟩, (i ∈ {1, 2, …, m}).

Then g(λ_{1}), …, g(λ_{m}) are the eigenvalues of g(P aP), and hence

tr[g(P aP)] = ∑_{i=1}^{m} g(λ_{i}) = ∑_{i=1}^{m} g(⟨ae_{i}, e_{i}⟩).  (4.5)

Since the trace on B(P (L)) is independent of choise of orthonormal basis for P (L), we have at the same time, that

tr[P g(a)P] = ∑_{i=1}^{m}⟨P g(a)P e_{i}, e_{i}⟩ = ∑_{i=1}^{m}⟨g(a)e_{i}, e_{i}⟩.  (4.6)

Comparing (4.5) and (4.6), we see that it suffices to show that ⟨g(a)e_{i}, e_{i}⟩ ≥ g(⟨ae_{i}, e_{i}⟩), for all i in {1, 2, …, m}. But for each i, this follows from the classical Jensen Inequality, applied to the distribution of a w.r.t. the state ⟨ · e_{i}, e_{i}⟩, i.e., the probability measure μ_{i} supported on sp(a), and satisfying that ⟨f(a)e_{i}, e_{i}⟩ = ∫_{sp(a)} f(t) dμ_{i}(t), for all functions f in C(sp(a)). This concludes the proof of (i).

(ii) By Stinespring’s Theorem, we may choose a Hilbert space L, a ∗-representation π: B → B(L) of B on L, and an embedding ι: ℂ^{m} → L of ℂ^{m} into L, such that

Ψ(b) = P_{K}π(b)P_{K},  (b ∈ B),  (4.7)

where K = ι(ℂ^{m}), and P_{K} is the orthogonal projection of L onto K. Moreover, the equality (4.7) is modulo the natural identifications associated with ι. Let tr_{K} denote the normalized trace on B(K). By application of (i), it follows then that

tr_{m}[g(Ψ(a))] = tr_{K}[g(P_{K}π(a)P_{K})] ≤ tr_{K}[P_{K}g(π(a))P_{K}]
= tr_{K}[P_{K}π(g(a))P_{K}] = tr_{m}[Ψ(g(a))],

and this proves (ii). ■

4.2 Lemma. Let V_{n}, n ∈ ℕ, be as in (4.3), and let λ_{max}(V_{n}) denote the largest eigenvalue of V_{n} (considered as an element of M_{dn}(ℂ)). Then for any ϵ in ]0, ∞[, we have that

∑_{n=1}^{∞} P (λ_{max}(V_{n}) ≥ (\sqrt{c} + 1)^{2} + ϵ) < ∞.  (4.8)

Proof. The proof proceeds along the same lines as the proof of [HT, Lemma 7.3]; the main difference being that in the present situation, we have to rely on the estimate obtained in Theorem 3.3. Consider first a fixed n in ℕ. We find then for any t in ]0, ∞[, that

P (λ_{max}(V_{n}) ≥ (\sqrt{c} + 1)^{2} + ϵ) = P ( exp (tλ_{max}(V_{n}) − t(\sqrt{c} + 1)^{2} − tϵ) ≥ 1)
≤ 𝔼[ exp (tλ_{max}(V_{n}) − t(\sqrt{c} + 1)^{2} − tϵ)]
= exp(−t(\sqrt{c} + 1)^{2} − tϵ) · 𝔼[λ_{max}( exp(tV_{n}))]
≤ exp(−t(\sqrt{c} + 1)^{2} − tϵ) · 𝔼[Tr_{dn}( exp(tV_{n}))],

(4.9)

where the last inequality follows by noting, that since exp(tV_{n}) is a positive dn × dn matrix, λ_{max}(exp(tV_{n})) ≤ Tr_{dn}(exp(tV_{n})). Note now, that since the mapping Φ ⊗ id_{n} is unital, completely positive, and since the function x ↦ e^{tx}: ℝ → ℝ is convex, it follows from Lemma 4.1(ii), that

tr_{dn}[ exp(tV_{n})] = tr_{dn}[ exp (t(Φ ⊗ id_{n})(S_{n}^{∗}S_{n}))] ≤ tr_{dn}[(Φ ⊗ id_{n})(exp(tS_{n}^{∗}S_{n}))]
= tr_{d} ⊗ tr_{n}[(Φ ⊗ id_{n})(exp(tS_{n}^{∗}S_{n}))] = ϕ ⊗ tr_{n}[ exp(tS_{n}^{∗}S_{n})],

(4.10)

where ϕ is the state tr_{d} ∘ Φ on A. Note here, that by Definition 3.1 and Theorem 3.3,

𝔼[ϕ ⊗ tr_{n}( exp(tS_{n}^{∗}S_{n}))] = ϕ ⊗ tr_{n}(𝔼[ exp(tS_{n}^{∗}S_{n})])

≤ exp (t(\sqrt{c} + 1)^{2} + \frac{t^{2}}{n} (c + 1)^{2}), (4.11)

for all t in ]0, \frac{n}{2c}].

Combining now (4.9)-(4.11), we get that for all t in ]0, \frac{n}{2c}],

P (λ_{max}(V_{n}) ≥(\sqrt{c} + 1)^{2} + ϵ)
≤ dn · exp(−t(\sqrt{c} + 1)^{2} − tϵ) · exp (t(\sqrt{c} + 1)^{2} + \frac{t^{2}}{n} (c + 1)^{2})
= dn · exp (t(\frac{t}{n}(c + 1)^{2} − ϵ)),

Now choose t = t_{n} = \frac{nϵ}{2(c+1)^{2}} , and note that t_{n} ∈ ]0, \frac{n}{2c}], as long as ϵ ≤ 1. Clearly it suffices to prove the lemma for such ϵ, so we assume that ϵ ≤ 1. It follows then that

P (λ_{max}(V_{n}) ≥ (\sqrt{c} + 1)^{2} + ϵ) ≤ dn · exp (t_{n}(\frac{t_{n}}{n} (c + 1)^{2} − ϵ)) = dn · exp (\frac{−nϵ^{2}}{4(c+1)^{2}} ).

Since this estimate holds for all n in ℕ, it follows immediately that (4.8) holds. ■

4.3 Proposition. Let V_{n}, n ∈ ℕ, be as in (4.3). We then have

lim sup_{n→∞} λ_{max}(V_{n}) ≤ (\sqrt{c} + 1)^{2}, almost surely.

Proof. It suffices to show, that for any ϵ from ]0, ∞[,

P ( lim sup_{n→∞} λ_{max}(V_{n}) ≤ (\sqrt{c} + 1)^{2} + ϵ) = 1,

and this will follow, if we show that

P (λ_{max}(V_{n}) ≤ (\sqrt{c} + 1)^{2} + ϵ, for all but finitely many n) = 1,

for all ϵ in ]0, ∞[. But this follows from the Borel-Cantelli Lemma (cf. [Bre, Lemma 3.14]) together with Lemma 4.2. ■

The next step is to replace V_{n} in Proposition 4.3 by S_{n}^{∗}S_{n} itself. This is where we need to assume that A is an exact C^{∗}-algebra. The key point in this step is the important result of E. Kirchberg that exactness implies nuclear embeddability (cf. [Ki2, Theorem 4.1] and [Was, Theorem 7.3]).

Let B be a unital C^{∗}-algebra. Recall then that an operator system in B is a subspace E of B, such that 111_{B} ∈ E and x^{∗} ∈ E for all x in E.

4.4 Proposition. Let B be a unital exact C^{∗}-algebra, and let E be a finite dimensional operator system in B. Then for any ϵ in ]0, ∞[, there exist d in ℕ and a unital completely positive mapping Φ: B → M_{d}(ℂ), such that

∥(Φ ⊗ id_{n})(x)∥ ≥ (1 − ϵ)∥x∥,

for all n in ℕ and all x in M_{n}(E).

Proof. Clearly we may assume that B is a unital C^{∗}-subalgebra of B(L) for some Hilbert space L. Let N denote the dimension of E. Then by Auerbach’s Lemma (cf. [LT, Proposition 1.c.3]), we may choose linear bases e_{1}, …, e_{N} of E and e_{1}^{∗}, …, e_{N}^{∗} of the dual space E^{∗}, such that

∥e_{i}∥ = ∥e_{i}^{∗}∥ = 1, and e_{i}^{∗}(e_{j}) = δ_{i,j},  (i, j ∈ {1, 2, …, N}).  (4.12)

Now since B is exact, and hence nuclear embeddable, there exist d in ℕ, and unital completely positive mappings Φ: B → M_{d}(ℂ) and Ψ: M_{d}(ℂ) → B(L), such that

∥Ψ(Φ(e_{i})) − e_{i}∥ ≤ \frac{ϵ}{N} ,  (i ∈ \{1, 2, …, N\}),  (4.13)

(cf. [Was, p. 60]). We show that this Φ has the property set out in the proposition. For this, it suffices to show that

∥(Ψ ∘ Φ − ι_{B})_{|E}∥_{cb} ≤ ϵ,  (4.14)

where ι_{B}: B → B(L) is the embedding of B into B(L). Indeed, knowing the validity of (4.14), we have for n in ℕ and x in M_{n}(E), that

∥x∥ ≤ ∥((Ψ ∘ Φ) ⊗ id_{n})(x) − x∥ + ∥((Ψ ∘ Φ) ⊗ id_{n})(x)∥ ≤ ϵ∥x∥ + ∥((Ψ ∘ Φ) ⊗ id_{n})(x)∥,

and hence that

(1 − ϵ)∥x∥ ≤ ∥((Ψ ∘ Φ) ⊗ id_{n})(x)∥ ≤ ∥(Φ ⊗ id_{n})(x)∥,

where the last inequality is due to the fact that Ψ, being unital completely positive, is a complete contraction.

To verify (4.14) note first, that for x in E, we have by (4.12),

x = ∑_{i=1}^{N} e_{i}^{∗}(x)e_{i},

and hence

Ψ ∘ Φ(x) − x = ∑_{i=1}^{N} e_{i}^{∗}(x)(Ψ ∘ Φ(e_{i}) − e_{i}) = ∑_{i=1}^{N} e_{i}^{∗}(x)f_{i},

where f_{i} = Ψ ∘ Φ(e_{i}) − e_{i}. Note that by (4.13), ∥f_{i}∥ ≤ \frac{ϵ}{N}, for all i in \{1, 2, …, N\}.

Consider now n in ℕ and x = (x_{rs})_{1≤r,s≤n} in M_{n}(E). We then have

((Ψ ∘ Φ) ⊗ id_{n})(x) − x = [(Ψ ∘ Φ)(x_{rs}) − x_{rs}]_{1≤r,s≤n}

= [∑_{i=1}^{N}e_{i}^{∗}(x_{rs})f_{i}]_{1≤r,s≤n}

= ∑_{i=1}^{N} ([e_{i}^{∗}(x_{rs})]_{1≤r,s≤n} · diag_{n}(f_{i}, …, f_{i})),

(4.15) where diag_{n}(f_{i}, …, f_{i}) is the n × n diagonal matrix with f_{i} in all the diagonal positions. Note here that by (4.12), ∥e_{i}^{∗}∥_{cb} = ∥e_{i}^{∗}∥ = 1, for all i (cf. [Pa, Proposition 3.7]). Consequently,

∥[e_{i}^{∗}(x_{rs})]_{1≤r,s≤n}∥ ≤ ∥e_{i}^{∗}∥_{cb} · ∥x∥ = ∥x∥, (i ∈ {1, 2, …, N}),

and using this in (4.15), we get that

∥((Ψ ∘ Φ) ⊗ id_{n})(x) − x∥ ≤ ∑_{i=1}^{N} ∥x∥ · ∥f_{i}∥ ≤ ∑_{i=1}^{N} ∥x∥\frac{ϵ}{N} = ϵ∥x∥,

which proves (4.14). ■

4.5 Theorem. Let a_{1}, …, a_{r} be elements of B(H, K), such that ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ ≤ c, and ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ ≤ 1, for some constant c in ]0, ∞[. Assume, in addition, that the C^{∗}- subalgebra A of B(H), generated by {a_{i}^{∗}a_{j} | i, j ∈ {1, 2, …, r}} ∪ {111_{B(H)}}, is exact. Consider furthermore, for each n in ℕ, independent elements Y_{1}^{(n)}, …, Y_{r}^{(n)} of GRM(n, n, \frac{1}{n}), and put S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}. We then have

lim sup_{n→∞} max [sp(S_{n}^{∗}S_{n})] ≤ (\sqrt{c} + 1)^{2}, almost surely.

Proof. It suffices to show, that for any ϵ from ]0, ∞[, the set

T_{ϵ} = \{ω ∈ Ω | lim sup_{n→∞} max [sp(S_{n}(ω)^{∗}S_{n}(ω))] ≤ \frac{1}{1−ϵ}(\sqrt{c} + 1)^{2}\},

has probability 1. So let ϵ from ]0, ∞[ be given, and put

E = span({111_{A}} ∪ {a_{i}^{∗}a_{j} | i, j ∈ {1, 2, …, r}}).

Note that x^{∗} ∈ E for all x in E, and that 111_{A} ∈ E. Hence E is a finite dimensional operator system in A. Since A is exact, it follows thus from Proposition 4.4, that we may choose d in ℕ and a completely positive mapping Φ: A → M_{d}(ℂ), such that

∥(Φ ⊗ id_{n})(x)∥ ≥ (1 − ϵ)∥x∥,  (n ∈ ℕ, x ∈ M_{n}(E)).  (4.16)

Now put

V_{n} = (Φ ⊗ id_{n})(S_{n}^{∗}S_{n}), (n ∈ ℕ),

and define furthermore

V = \{ω ∈ Ω | lim sup_{n→∞} ∥V_{n}(ω)∥ ≤ (\sqrt{c} + 1)^{2}\}.

By Proposition 4.3, P (V) = 1, and hence it suffices to show that T_{ϵ} ⊇ V. But if ω ∈ V, it follows from (4.16) that

lim sup_{n→∞} ∥S_{n}(ω)^{∗}S_{n}(ω)∥ ≤ (1 − ϵ)^{−1} lim sup_{n→∞} ∥V_{n}(ω)∥ ≤ (1 − ϵ)^{−1}(\sqrt{c} + 1)^{2},

which shows that ω ∈ T_{ϵ}. This concludes the proof. ■

4.6 Corollary. Let a_{1}, …, a_{r} be elements of an exact C^{∗}-algebra A, and let, for each n in ℕ, Y_{1}^{(n)}, …, Y_{r}^{(n)} be independent elements of GRM(n, n, \frac{1}{n}). Then

lim sup_{n→∞} ∥ ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}∥ ≤ ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥^{\frac{1}{2}} + ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥^{\frac{1}{2}} , almost surely.

Proof. We may assume that not all a_{i} are zero. Put γ = ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥ > 0 and δ = ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥ > 0. We may assume that A ⊆ B(H) for some Hilbert space H. Then the unital C^{∗}-algebra Ã = C^{∗}(A,111_{B(H)}) is also exact, and hence so is every C^{∗}-subalgebra of Ã (cf. [Ki1] and [Was, 2.5.2]). Therefore Corollary 4.6 follows by applying Theorem 4.5 to a′_{i} = \frac{1}{\sqrt{δ}}a_{i}, i = 1, …, r, and c = \frac{γ}{δ}. ■

Regarding the corollary above, consider arbitrary elements a_{1}, …, a_{r} of an arbitrary C^{∗}-algebra A, and let {y_{1}, …, y_{r}} be a circular (or semi-circular) system in some C^{∗}- probability space (B, ψ) (cf. [Vo2]), and normalized so that ψ(y_{i}^{∗}y_{i}) = 1, i = 1, 2, …, r. In [HP, Proof of Proposition 4.8], G. Pisier and the first named author showed, that in this setting, the following inequality holds:

∥ ∑_{i=1}^{r} a_{i} ⊗ y_{i}∥ ≤ 2 max \{∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥^{\frac{1}{2}} , ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥^{\frac{1}{2}} \}.  (4.17)

In [HP, Proof of Proposition 4.8], the factor 2 on the right hand side of (4.17) is missing, but this is due to a different choise of normalization of semi-circular and circular families. By application of [Haa, Section 1], it is not hard to strengthen (4.17) to the inequality

∥ ∑_{i=1}^{r} a_{i} ⊗ y_{i}∥ ≤ ∥ ∑_{i=1}^{r} a_{i}^{∗}a_{i}∥^{\frac{1}{2}} + ∥ ∑_{i=1}^{r} a_{i}a_{i}^{∗}∥^{\frac{1}{2}} ,  (4.18)

both for semi-circular and circular systems. Since independent elements Y_{1}^{(n)}, …, Y_{r}^{(n)} of GRM(n, n, \frac{1}{n}) can be considered as a random matrix model for the circular system {y_{1}, …, y_{r}}, in the sense of [Vo1, Theorem 2.2], we should thus consider Corollary 4.6 as a random matrix version of (4.18). However, the random matrix version holds only under the assumption that the C^{∗}-algebra A be exact. In fact, we shall spend the remaining part of this section, showing that the assumption in Theorem 4.5 that the C^{∗}-algebra A be exact, can not be omitted. We start with two lemmas, the first of which is a slightly strengthened version of [HT, Theorem 7.4] (which, in turn, is a special case of a theorem of Wachter (cf. [Wac])).

4.7 Lemma. Let c be a positive number, and let (m_{n}) be a sequence of positive integers, such that \frac{m_{n}}{n} → c as n → ∞. Let furthermore (Y_{n}) be a sequence of random matrices, such that for each n in ℕ, Y_{n} ∈ GRM(m_{n}, n, \frac{1}{n}). Then for any continuous function f: [0, ∞[ → ℂ, we have that

lim_{n→∞} tr_{n}[f(Y_{n}^{∗}Y_{n})] = ∫_{0}^{b} f(x) dμ_{c}(x),  almost surely,  (4.19)

where b = (\sqrt{c} + 1)^{2} and μ is the measure introduced in (3.13).

Proof. By splitting f in its real and imaginary parts, it is clear, that we may assume that f is a real valued continuous function on [0, ∞[. We note next, that it follows from [HT, Theorem 7.4] and the definition of weak convergence (cf. [HT, Definition 2.2]), that (4.19) holds for all continuous bounded functions f: [0, ∞[ → ℝ. Thus, our objective is to pass from bounded to unbounded continuous functions, and the key to this, is the fact (cf. [HT, Theorem 7.1]), that

lim_{n→∞} ∥Y_{n}^{∗}Y_{n}∥ = (\sqrt{c} + 1)^{2},  almost surely.  (4.20)

Indeed, it follows from (4.20), that (for example)

P ( ∥Y_{n}^{∗}Y_{n}∥ ≤ (\sqrt{c} + 1)^{2} + 1, for all but finitely many n) = 1,

and hence, given any ϵ in ]0, ∞[, we may choose N in ℕ, such that

P (F_{N}) ≥ 1 − ϵ,

where

F_{N} = \{ω ∈ Ω | ∥Y_{n}(ω)^{∗}Y_{n}(ω)∥ ≤ (\sqrt{c} + 1)^{2} + 1, whenever n ≥ N\}.

Now, given a continuous function f: [0, ∞[ → ℝ, let f_{1} : [0, ∞[ → ℝ be an arbitrary continuous function, satisfying that f_{1} = f on [0, (\sqrt{c} + 1)^{2} + 1], and that supp(f) is compact. Then for any ω in F_{N}, we have that

f_{1}(Y_{n}(ω)^{∗}Y_{n}(ω)) = f(Y_{n}(ω)^{∗}Y_{n}(ω)), whenever n ≥ N,

and hence, since f_{1} is bounded,

lim_{n→∞} tr_{n}[f(Y_{n}(ω)^{∗}Y_{n}(ω))] = lim_{n→∞} tr_{n}[f_{1}(Y_{n}(ω)^{∗}Y_{n}(ω))] = ∫_{a}^{b} f_{1}(x) dμ_{c}(x)
= ∫_{a}^{b} f(x) dμ_{c}(x).

It follows thus, that

P ( lim_{n→∞} tr_{n}[f(Y_{n}^{∗}Y_{n})] = ∫_{a}^{b}f(x) dμ_{c}(x)) ≥ P (F_{N}) ≥ 1 − ϵ,

and since this holds for any ϵ in ]0, ∞[, we obtain the desired conclusion. ■

Next, we shall study the polar decomposition of Gaussian random matrices. Let n be a positive integer and let Y be an element of GRM(n, n, \frac{1}{n}), defined on (Ω, F, P). Furthermore, let U_{n} denote the unitary group of M_{n}(ℂ).

By a measurable unitary sign for Y, we mean a random matrix U: Ω → U_{n}, such that for almost all ω in Ω, the polar-decomposition of Y (ω) is given by:

Y (ω) = U(ω)|Y (ω)|,

where, as usual, |Y (ω)| = [Y (ω)^{∗}Y (ω)]^{\frac{1}{2}} . To see that such measurable unitary signs do exist, we note first that by [HT, Theorem 5.2], Y (ω) is invertible for almost all ω. Thus, for example the random matrix U: Ω → U_{n} given by

U(ω) = \begin{cases}Y (ω)[Y (ω)^{∗}Y (ω)]^{− \frac{1}{2}} , & if Y (ω) is invertible, \\ 111_{n}, & otherwise, \end{cases}

is a measurable unitary sign for Y.

4.8 Lemma. For each n in ℕ, let Y_{1}^{(n)}, …, Y_{r}^{(n)} be (not necessarily independent) random matrices in GRM(n, n, \frac{1}{n}), and let U_{1}^{(n)}, …, U_{r}^{(n)} be measurable unitary signs for Y_{1}^{(n)}, …, Y_{r}^{(n)}, respectively. Furthermore, let \overline{U}_{1}^{(n)}, …, \overline{U}_{r}^{(n)}, denote the complex conjugated matrices of U_{1}^{(n)}, …, U_{r}^{(n)}. We then have

lim inf_{n→∞} ∥ ∑_{i=1}^{r} \overline{U}_{i}^{(n)} ⊗ Y_{i}^{(n)}∥ ≥ \frac{8}{3π} · r, almost surely.

Proof. Let (e_{1}, …, e_{n}) be the usual orthonormal basis for ℂ^{n}, and consider then the unit vector ξ = \frac{1}{\sqrt{n}} ∑_{i=1}^{n} e_{i} ⊗ e_{i} in ℂ^{n} ⊗ ℂ^{n}. Note then that for any A = (a_{jk}) and B = (b_{jk}) in M_{n}(ℂ), we have that

\begin{aligned}⟨(A ⊗ B)ξ, ξ⟩ & = \frac{1}{n} ∑_{j,k=1}^{n}⟨(A ⊗ B)(e_{j} ⊗ e_{j}), e_{k} ⊗ e_{k}⟩ = \frac{1}{n} ∑_{j,k=1}^{n}⟨Ae_{j}, e_{k}⟩ · ⟨Be_{j}, e_{k}⟩ \\ & = \frac{1}{n} ∑_{j,k=1}^{n}a_{kj}b_{kj} = tr_{n}(AB^{t}) = tr_{n}(A^{t}B). \end{aligned}

It follows thus, that

\begin{aligned}∥ ∑_{i=1}^{r} \overline{U}_{i}^{(n)} ⊗ Y_{i}^{(n)}∥ & ≥ |⟨( ∑_{i=1}^{r} \overline{U}_{i}^{(n)} ⊗ Y_{i}^{(n)})ξ, ξ⟩| = | ∑_{i=1}^{r} tr_{n}[(U_{i}^{(n)})^{∗}Y_{i}^{(n)}]| \\ & = ∑_{i=1}^{r} tr_{n}(|Y_{i}^{(n)}|), \end{aligned}  (4.21)

where the last equation holds almost surely. By Lemma 4.7, we have for all i in {1, …, r}, that

lim_{n→∞} tr_{n}(|Y_{i}^{(n)}|) = ∫_{0}^{4} \sqrt{x} dμ_{1}(x), almost surely,

and combining this with (4.21), it follows that

lim inf_{n→∞} ∥ ∑_{i=1}^{r} \overline{U}_{i}^{(n)} ⊗ Y_{i}^{(n)}∥ ≥ r ∫_{0}^{4} \sqrt{x} dμ_{1}(x), almost surely.

We note finally that

∫_{0}^{4} \sqrt{x} dμ_{1}(x) = ∫_{0}^{4} \sqrt{x} · ^{\frac{\sqrt{x(4−x)}}{2πx}} dx = \frac{1}{2π}∫_{0}^{4}\sqrt{4 − x} dx = \frac{8}{3π},

and this concludes the proof. ■

We are now ready to give an example where the conclusion of Theorem 4.5 fails, due to lack of exactness of the C^{∗}-algebra A. Consider a fixed positive integer r, greater than or equal to 2, and let 𝔽_{r} denote the free group on r generators. Let g_{1}, …, g_{r} denote the generators of 𝔽_{r}, and let C^{∗}(𝔽_{r}) denote the full C^{∗}-algebra associated to 𝔽_{r}. Recall that there is a canonical unitary representation u_{𝔽_{r}}: 𝔽_{r} → C^{∗}(𝔽_{r}), and that the pair (C^{∗}(𝔽_{r}), u_{𝔽_{r}}) is characterized (up to ∗-isomorphism) by the universial property, that given any unital C^{∗}-algebra B and any unitary representation u: 𝔽_{r} → B, there exists a unique unital ∗-homomorphism Φ_{u}: C^{∗}(𝔽_{r}) → B, such that the following diagram commutes:

\xymatrix{𝔽_{r} \ar[r]^{u_{𝔽r}} \ar[d]_{u} & C^{∗}(𝔽_{r}) \ar[dl]^{Φ_{u}} \\ B}

It is well-known (cf. [Was, Corollary 3.7]) that C^{∗}(𝔽_{r}) is not exact. We let u_{1}, …, u_{r} be the canonical unitaries in C^{∗}(𝔽_{r}) associated to g_{1}, …, g_{r} respectively, i.e., u_{i} = u_{𝔽_{r}}(g_{i}), i = 1, …, r. We then define

a_{i} = \frac{1}{\sqrt{r}}u_{i},  (i ∈ \{1, …, r\}).  (4.22)

Then clearly,

∑_{i=1}^{r} a_{i}^{∗}a_{i} = ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{C^{∗}(𝔽_{r})}.  (4.23)

Consider now, in addition, for each n in ℕ, independent elements Y_{1}^{(n)}, …, Y_{r}^{(n)} of GRM(n, n, \frac{1}{n}), and define

S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)},  (n ∈ ℕ).  (4.24)

We then have the following

4.9 Proposition. With a_{1}, …, a_{r} and S_{n}, n ∈ ℕ, as introduced in (4.22) and (4.24), we have that

(i) lim inf_{n→∞} ∥S_{n}^{∗}S_{n}∥ ≥ (\frac{8}{3π})^{2} · r, almost surely.

(ii) The conclusion of Theorem 4.5 does not hold for these a_{1}, …, a_{r}, whenever r ≥ 6.

In particular, the assumption in Theorem 4.5, that A be exact, can not, in general, be omitted.

Proof. (i) For each positive integer n, choose measurable unitary signs U_{1}^{(n)}, …, U_{r}^{(n)} for Y_{1}^{(n)}, …, Y_{r}^{(n)} respectively, and let \overline{U}_{1}^{(n)}, …, \overline{U}_{r}^{(n)} denote the complex conjugated matrices of U_{1}^{(n)}, …, U_{r}^{(n)}. Since 𝔽_{r} is the group free product of r copies of ℤ, it follows that for each ω in Ω and each n in ℕ, there exists a unitary representation u_{ω}^{(n)} : 𝔽_{r} → M_{n}(ℂ), such that

u_{ω}^{(n)}(g_{i}) = \overline{U}_{i}^{(n)}(ω), (i ∈ \{1, …, r\}).

By the universial property of C^{∗}(𝔽_{r}) it follows then, that for each ω in Ω and each n in ℕ, we may choose a ∗-homomorphism Φ_{ω}^{(n)} : C^{∗}(𝔽_{r}) → M_{n}(ℂ), such that

Φ_{ω}^{(n)}(u_{i}) = \overline{U}_{i}^{(n)}(ω), (i ∈ \{1, …, r\}).

For each ω in Ω and each n in ℕ, note now that

∥ ∑_{i=1}^{r} u_{i} ⊗ Y_{i}^{(n)}(ω)∥ ≥ ∥(Φ_{ω}^{(n)} ⊗ id_{n})( ∑_{i=1}^{r} u_{i} ⊗ Y_{i}^{(n)}(ω))∥ = ∥ ∑_{i=1}^{r} \overline{U}_{i}^{(n)}(ω) ⊗ Y_{i}^{(n)}(ω)∥.

Applying then Lemma 4.8, it follows that

lim inf_{n→∞} ∥ ∑_{i=1}^{r} u_{i} ⊗ Y_{i}^{(n)}∥ ≥ \frac{8}{3π} · r, almost surely,

and hence that

lim inf_{n→∞} ∥ ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}∥ ≥ \frac{8}{3π} · \sqrt{r}, almost surely.

Since ∥S_{n}^{∗}S_{n}∥ = ∥S_{n}∥^{2}, we get the desired formula.

(ii) By (4.23), a_{1}, …, a_{r} introduced in (4.22) satisfy condition (4.1) in the case c = 1. Thus, if the conclusion of Theorem 4.5 were to hold for these a_{1}, …, a_{r}, it would mean that

lim sup_{n→∞} ∥ ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}∥ ≤ 2, almost surely.

However, Proposition 4.9 shows that

lim inf_{n→∞} ∥ ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}∥ ≥ (\frac{8}{3π}) · \sqrt{r}, almost surely,

and thus the conclusion of Theorem 4.5 breaks down, for c = 1, whenever r > (\frac{3π}{4})^{2} ≈ 5.55, i.e., for r ≥ 6. ■

5 A New Combinatorial Expression for 𝔼[(S^{∗}S)^{p}]

Throughout this section, we consider elements a_{1}, …, a_{r} of B(H, K), where H and K are Hilbert spaces. In Section 2 we proved that if Y_{1}, …, Y_{r} are independent random matrices in GRM(n, n, \frac{1}{n}), and we put S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}, then

𝔼[(S^{∗}S)^{p}] = ( ∑_{π∈S_{p}} n^{−2σ(π̂)} · ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) ⊗ 111_{n}.  (5.1)

In this section, we shall assume that a_{1}, …, a_{r} satisfy the condition

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)}, and ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)},  (5.2)

for some number c in ]0, ∞[. Under this assumption, and by application of the method of “reductions of permutations”, introduced in Section 1, we show that 𝔼[(S^{∗}S)^{p}] can be expressed as a constant plus a linear combination of the sums:

∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}}), (q = 2, …, p),

where S_{q}^{irr}, as in Section 1, denotes the set of permutations ρ in S_{q} for which ρ̂ is irreducible in the sence of Definition 1.16.

5.1 Lemma. Let a_{1}, …, a_{r} be elements of B(H, K), and assume that (5.2) holds. Let p be a positive integer, greater than or equal to 2, let π be a permutation in S_{p} \ S_{p}^{irr}, and let π_{0} be the permutation in S_{p−1} obtained by cancellation of a pair (e, e + 1) for π̂ (cf. Definition 1.18). We then have

(i) If e is odd, then k(π̂_{0}) = k(π̂) − 1, and

∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} = c · ( ∑_{1≤i_{1},... ,i_{p−1}≤r} a_{i_{1}}^{∗}a_{i_{π_{0}(1)}} ⋯ a_{i_{p−1}}^{∗}a_{i_{π_{0}(p−1)}}).  (5.3)

(ii) If e is even, then k(π̂_{0}) = k(π̂), and

∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} = ∑_{1≤i_{1},... ,i_{p−1}≤r} a_{i_{1}}^{∗}a_{i_{π_{0}(1)}} ⋯ a_{i_{p−1}}^{∗}a_{i_{π_{0}(p−1)}}.  (5.4)

Proof. (i) Assume that e is odd. Then k(π̂_{0}) = k(π̂) − 1 by Proposition 1.22. Moreover, (e, e + 1) is of the form (2j − 1, 2j) for some j in {1, 2, …, p}, and therefore π(j) = j (cf. Definition 1.15). Hence, the index i_{j} occur only at the 2j − 1’th and the 2j’th factor in the product a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}, and therefore the sum on the left hand side of (5.3) is equal to

∑_{1≤i_{1},... ,i_{j−1},i_{j+1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{π(j−1)}}( ∑_{i_{j}=1}^{r} a_{i_{j}}^{∗}a_{i_{j}})a_{i_{j+1}}^{∗} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}},

which by (5.2) is equal to

c · ( ∑_{1≤i_{1},... ,i_{j−1},i_{j+1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{π(j−1)}}a_{i_{j+1}}^{∗} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}).  (5.5)

Note here, that if we relabel the indices i_{j+1}, …, i_{p} by i_{j}, …, i_{p−1}, then it follows from Remark 1.19(a), that (5.5) is equal to

c · ( ∑_{1≤i_{1},... ,i_{p−1}≤r} a_{i_{1}}^{∗}a_{i_{π_{0}(1)}} ⋯ a_{i_{p−1}}^{∗}a_{i_{π_{0}(p−1)}}),

and this proves (5.3).

(ii) Assume that e is even. Then k(π̂_{0}) = k(π̂) by Proposition 1.22, and (e, e + 1) = (2j, 2j + 1), for some j in {1, 2, …, p − 1}, so that π(j) = j + 1 (c.f. Definition 1.15). Hence, the left hand side of (5.4) is equal to

∑_{1≤i_{1},... ,i_{j},i_{j+2},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{j}}^{∗}( ∑_{i_{j+1}=1}^{r} a_{i_{j+1}}a_{i_{j+1}}^{∗})a_{i_{π(j+1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}.  (5.6)

Here, ∑_{i_{j+1}=1}^{r} a_{i_{j+1}}a_{i_{j+1}}^{∗} = 111_{B(K)}, by (5.2), and proceeding then as in the proof of (i), we obtain by Remark 1.19(b) (after relabeling i_{j+2}, …, i_{p} by i_{j+1}, …, i_{p−1}), that (5.6) is equal to

∑_{1≤i_{1},... ,i_{p−1}≤r} a_{i_{1}}^{∗}a_{i_{π_{0}(1)}} ⋯ a_{i_{p−1}}^{∗}a_{i_{π_{0}(p−1)}}.

This proves (5.4) ■

Recall that for p in ℕ, S_{p}^{nc} denotes the set of permutations π in S_{p}, for which the permutation π̂ is non-crossing in the sense of Definition 1.14.

5.2 Lemma. Let a_{1}, …, a_{r} be elements of B(H, K), such that (5.2) holds, let p be a positive integer, and let π be a permutation in S_{p}^{nc}. Then

∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} = c^{k(π̂)}111_{B(H)},  (5.7)

and

∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}a_{i_{π(1)}}^{∗} ⋯ a_{i_{p}}a_{i_{π(p)}}^{∗} = c^{l(π̂)−1}111_{B(K)}.  (5.8)

Proof. We start by proving (5.7); proceeding by induction on p. The case p = 1 is clear from (5.2). Assume now that p ≥ 2, and that (5.7) holds for p − 1 instead of p, and all permutations in S_{p−1}^{nc}. Consider then a permutation π from S_{p}^{nc}, and recall from Lemma 1.17 that π̂ has a pair of neighbours (e, e + 1). Let π_{0} be the permutation in S_{p−1} obtained by cancellation of this pair. Then by Lemma 1.20, π_{0} ∈ S_{p−1}^{nc}, and hence by the induction hypothesis,

∑_{1≤i_{1},... ,i_{p−1}≤r} a_{i_{1}}^{∗}a_{i_{π_{0}(1)}} ⋯ a_{i_{p−1}}^{∗}a_{i_{π_{0}(p−1)}} = c^{k(π̂_{0})}111_{B(H)}.  (5.9)

But by Lemma 5.1, (5.9) implies (5.7), both when e is odd, and when e is even. This completes the proof of (5.7).

To prove (5.8), we put b_{i} = \frac{1}{\sqrt{c}}a_{i}^{∗}, i = 1, 2, ⋯, r. Then

∑_{i=1}^{r} b_{i}^{∗}b_{i} = c^{−1}111_{B(K)}, and ∑_{i=1}^{r} b_{i}b_{i}^{∗} = 111_{B(H)}.

Applying then (5.7), with c replaced by c^{−1}, it follows that

∑_{1≤i_{1},... ,i_{p}≤r} b_{i_{1}}^{∗}b_{i_{π(1)}} ⋯ b_{i_{p}}^{∗}b_{i_{π(p)}} = c^{−k(π̂)}111_{B(K)},

i.e., that

∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}a_{i_{π(1)}}^{∗} ⋯ a_{i_{p}}a_{i_{π(p)}}^{∗} = c^{p−k(π̂)}111_{B(K)}.

Recall finally, that since π̂ is non-crossing, k(π̂) + l(π̂) = p + 1 (cf. Corollary 1.24), and hence it follows that (5.8) holds. ■

As in Section 3, for any c in ]0, ∞[, μ_{c} denotes the probability measure on [0, ∞[, given by

μ_{c} = max\{1 − c, 0\}δ_{0} + \frac{\sqrt{(x − a)(b − x)}}{2πx} · 1_{[a,b]}(x) · dx,

where a = (\sqrt{c} − 1)^{2}, b = (\sqrt{c} + 1)^{2} and δ_{0} is the Dirac measure at 0. Recall from [OP] or [HT, Remark 6.8], that the moments of μ_{c} are given by

∫_{0}^{∞} x^{p} dμ_{c}(x) = \frac{1}{p} ∑_{j=1}^{p}\binom{p}{j}\binom{p}{j−1}c^{j},  (p ∈ ℕ).  (5.10)

5.3 Lemma. For any positive integer p, we have

∑_{π∈S_{p}^{nc}} c^{k(π̂)} = \frac{1}{p} ∑_{j=1}^{p}\binom{p}{j}\binom{p}{j−1}c^{j},  (5.11)

and

∑_{π∈S_{p}^{nc}} c^{l(π̂)−1} = \frac{1}{p} ∑_{j=1}^{p}\binom{p}{j}\binom{p}{j−1}c^{j−1}.  (5.12)

Proof. To prove (5.11), recall from Corollary 1.12, that for B in GRM(m, n, 1), we have that

𝔼 ∘ Tr_{n}[(B^{∗}B)^{p}] = ∑_{π∈S_{p}} m^{k(π̂)}n^{l(π̂)}.

Hence, for Y in GRM(m, n, \frac{1}{n}),

𝔼 ∘ tr_{n}[(Y^{∗}Y)^{p}] = n^{−p−1} ∑_{π∈S_{p}} m^{k(π̂)}n^{l(π̂)} = ∑_{π∈S_{p}} n^{−2σ(π̂)}(\frac{m}{n})^{k(π̂)},  (5.13)

where we have used that σ(π̂) = \frac{1}{2}(p + 1 − k(π̂) − l(π̂)). Consider now a sequence (m_{n}) of positive integers, such that \frac{m_{n}}{n} → c as n → ∞, and for each n in ℕ, let Y_{n} be an element of GRM(m_{n}, n, \frac{1}{n}). It follows then from (5.13), that

lim_{n→∞} 𝔼 ∘ tr_{n}[(Y^{∗}Y)^{p}] = ∑_{\begin{matrix}π∈S_{p} \\ σ(π̂)=0 \end{matrix}} c^{k(π̂)} = ∑_{π∈S_{p}^{nc}} c^{k(π̂)},  (5.14)

where the last equality follows from Corollary 1.24. On the other hand, it follows from [HT, Theorem 6.7(ii)] and (5.10), that

lim_{n→∞} 𝔼 ∘ tr_{n}[(Y^{∗}Y)^{p}] = ∫_{0}^{∞} x^{p} dμ_{c}(x) = \frac{1}{p} ∑_{j=1}^{p}\binom{p}{j}\binom{p}{j−1}c^{j}.  (5.15)

Combining (5.14) and (5.15), we obtain (5.11).

To prove (5.12), we use, again, that k(π̂) + l(π̂) = p + 1 for all π in S_{p}^{nc}. It follows thus, that

∑_{π∈S_{p}^{nc}} c^{l(π̂)−1} = c^{p} ∑_{π∈S_{p}^{nc}} c^{−k(π̂)}.  (5.16)

But by (5.11) (with c replaced by c^{−1}), the right hand side of (5.16) is equal to

\frac{1}{p} ∑_{j=1}^{p}\binom{p}{j}\binom{p}{j−1}c^{p−j}.  (5.17)

Substituting finally j with p + 1 − j in (5.17), we obtain (5.12). ■

5.4 Corollary. Let a_{1}, …, a_{r} be elements of B(H, K), such that (5.2) holds. Then for any p in ℕ, we have that

(i)  ∑_{π∈S_{p}^{nc}} ( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) = [\frac{1}{p} ∑_{j=1}^{p}\binom{p}{j}\binom{p}{j−1}c^{j}] · 111_{B(H)},

and

(ii)  ∑_{π∈S_{p}^{nc}} ( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}a_{i_{π(1)}}^{∗} ⋯ a_{i_{p}}a_{i_{π(p)}}^{∗}) = [\frac{1}{p} ∑_{j=1}^{p}\binom{p}{j}\binom{p}{j−1}c^{j−1}] · 111_{B(K)}.

Proof. Combine Lemma 5.2 and Lemma 5.3. ■

5.5 Definition. (a) A subset I of ℤ is called an interval of integers, if it is the form

I = {α, α + 1, …, β},

for some α, β in ℤ, such that α ≤ β.

(b) Let p be a positive integer, let π be a permutation in S_{p}, and let I be an interval of integers, such that I ⊆ {1, 2, …, 2p}. We say then that the restriction π̂_{|I} of π̂ to I is non-crossing, if π̂(I) = I, and π̂ has no crossing (a, b, c, d) where a, b, c, d ∈ I. In this case, we refer to I as a non-crossing interval of integers for π̂. □

5.6 Remark. Let p be a positive integer, let π be a permutation in S_{p} and let I be an interval of integers, such that I ⊆ {1, 2, …, 2p} and π̂(I) = I. Since π̂^{2} = id and π̂ has no fixed points, it follows then, that card(I) is an even number. Put t = \frac{1}{2}card(I), and consider the unique order preserving bijection φ: {1, 2, …, 2t} → I of {1, 2, …, 2t} onto I (i.e., φ(j) = min(I)−1+j, for all j in {1, 2, …, 2t}). It is clear then, that the mapping φ^{−1} ∘ (π̂_{|I}) ∘ φ is a permutation of {1, 2, …, 2t}, and that we may choose a (unique) permutation π_{1} in S_{t}, such that

π̂_{1} = φ^{−1} ∘ (π̂_{|I}) ∘ φ,  (5.18)

(cf. Remark 1.7(a)). It is clear too, that the restriction π̂_{|I} of π̂ to I is non-crossing in the sense of Definition 5.5, if and only if π̂_{1} is a non-crossing permutation in the usual sense (cf. Definition 1.14). □

5.7 Lemma. Let p be a positive integer, and let π be a permutation in S_{p}.

(i) If I is an interval of integers such that I ⊆ {1, 2, …, 2p} and π̂_{|I} is non-crossing, then there exists e in I, such that e + 1 ∈ I and π̂(e) = e + 1.

(ii) If π ∈ S_{p}^{irr}, then π̂ has no non-crossing interval of integers.

Proof. (i) Assume that I ⊆ \{1, 2, …, 2p\} and that π̂_{|I} is non-crossing. Put t = \frac{1}{2}card(I), let φ be the order preserving bijection of {1, 2, …, 2t} onto I, and let π_{1} be the permutation in S_{t} given by (5.18). Then π_{1} ∈ S_{t}^{nc}, and hence π̂_{1} has a pair of neighbours (e′, e′ +1) by Lemma 1.17. Putting e = φ(e′), it follows that e + 1 = π̂(e) ∈ I, and this proves (i).

(ii) This follows immediately from (i). ■

5.8 Lemma. Let p be a positive integer, and let π be a permutation in S_{p}, such that π̂ is reducible. Consider furthermore a family (I_{λ})_{λ∈Λ} of intervals of integers, such that I_{λ} ⊆ {1, 2, …, 2p} for all λ, and such that the union I = ∪_{λ∈Λ}I_{λ} is again an interval of integers. If each I_{λ} is a non-crossing interval of integers for π̂, then so is I.

Proof. Assume that each I_{λ} is a non-crossing interval of integers for π̂. Then π̂(I_{λ}) = I_{λ} for all λ, and hence also π̂(I) = I. Assume then that I contains a crossing for π̂, i.e., that there exist a, b, c, d in I, such that a < b < c < d and π̂(a) = c, π̂(b) = d. Choose λ in Λ such that a ∈ I_{λ}. Then c = π̂(a) ∈ I_{λ}, and since I_{λ} is an interval of integers, also b ∈ I_{λ}. But then d = π̂(b) ∈ I_{λ} too, and hence (a, b, c, d) is a crossing for π̂ contained in I_{λ}; a contradiction. Therefore I too is a non-crossing interval of integers for π̂. ■

5.9 Definition. Let p be a positive integer and let π be a permutation in S_{p}. By J (π̂) we denote then the family of all non-crossing intervals of integers for π̂. Moreover, we put

NC(π̂) = ⋃_{I∈J (π̂)} I,  (5.19)
IRR(π̂) = {1, 2, …, 2p} \ NC(π̂).  (5.20)

We refer to NC(π̂) (respectively IRR(π̂)) as the non-crossing set (respectively irreducible set) for π̂. □

5.10 Lemma. Let p be a positive integer and let π be a permutation in S_{p}. We then have

(i) NC(π̂) = {1, 2, …, 2p} if and only if π̂ is non-crossing.

(ii) NC(π̂) = ∅ if and only if π̂ is irreducible.

Proof. (i) If NC(π̂) = {1, 2, …, 2p}, then is follows from Lemma 5.8, that π̂ is noncrossing. If, conversely, π̂ is non-crossing, then {1, 2, …, 2p} ∈ J (π̂), and hence NC(π̂) = {1, 2, …, 2p}.

(ii) If NC(π̂) = ∅, then for any j in {1, 2, …, 2p−1}, {j, j +1} can not be a non-crossing interval of integers for π̂. Hence π̂(j) ≠ j + 1 for all j in {1, 2, …, 2p − 1}, which means that π̂ is irreducible. If, conversely, π̂ is irreducible, then J (π̂) = ∅ by Lemma 5.7(ii), and hence also NC(π̂) = ∅. ■

5.11 Proposition. Let p be a postitive integer, let π be a permutation in S_{p}, and assume that π̂ has a crossing. Then the set IRR(π̂) is of the form

IRR(π̂) = {s_{1}, s_{2}, …, s_{2q}},

where q ∈ {1, …, p}, and 1 ≤ s_{1} < s_{2} < ⋯ < s_{2q} ≤ 2p. Moreover, s_{1}, s_{2}, …, s_{2q} have the following properties:

(i) The set {s_{1}, s_{2}, …, s_{2q}} is π̂-invariant and π̂(s_{i}) ≠ s_{i+1}, for all i in {1, 2, …, 2q − 1}.

(ii) If we put s_{0} = 0 and s_{2q+1} = 2p + 1, then for each i in {0, 1, …, 2q}, the set

I_{i} = ]s_{i}, s_{i+1}[ ∩ ℤ

is either the empty set or a non-crossing interval of integers for π̂.

Proof. By Definition 5.5(b), each I in J (π̂) is π̂-invariant. Therefore NC(π̂) is π̂-invariant too, and hence so is IRR(π̂). Since π̂^{2} = id and π̂ has no fixed points, it follows that card(IRR(π̂)) = 2q for some q in {0, 1, …, p}, and since π̂ has a crossing, Lemma 5.10(i) shows that q ≥ 1. Thus, we may write IRR(π̂) in the form {s_{1}, s_{2}, …, s_{2q}}, where s_{1} < s_{2} < ⋯ < s_{2q}, and it remains to show that these s_{1}, s_{2}, …, s_{2q} satisfy (i) and (ii).

We start by proving (ii). For all I from J (π̂), I ∩ {s_{1}, s_{2}, …, s_{2q}} = ∅, and hence each such I is contained in one of the sets I_{i} = ]s_{i}, s_{i+1}[ ∩ ℤ, i = 0, 1, …, 2q. Therefore

J (π̂) = ⋃_{i=0}^{2q} J_{i}(π̂),  (5.21)

where J_{i}(π̂) = {I ∈ J (π̂) | I ⊆ I_{i}}, for all i in {0, 1, …, 2q}. Note here that

⋃_{I∈J_{i}(π̂)} I ⊆ I_{i},  (i ∈ {0, 1, …, 2q}),  (5.22)

and that

⋃_{I∈J (π̂)} I = NC(π̂) = {1, 2, …, 2p} \ IRR(π̂) = ⋃_{i=0}^{2q} I_{i}.  (5.23)

Combining (5.21)-(5.23), it follows that we actually have equality in (5.22), i.e.,

⋃_{I∈J_{i}(π̂)} I = I_{i},  (i ∈ {0, 1, …, 2q}).  (5.24)

Since each I_{i} is either empty or an interval of integers, (ii) follows now by combining (5.24) with Lemma 5.8.

It remains to prove (i). We already noted (and used) that IRR(π̂) is π̂-invariant. Assume then that π̂(s_{i}) = s_{i+1} for some i in {1, …, 2q − 1}. Then, by (ii), the set

Ĩ_{i} = {s_{i}} ∪ I_{i} ∪ {s_{i+1}},

is a non-crossing interval of integers for π̂. But this contradicts that s_{i} ∉ NC(π̂), and hence we have proved (i). ■

We prove next the following converse of Proposition 5.11.

5.12 Proposition. Let p be a positive integer, let π be a permutation in S_{p}, and assume that there exist q in {1, …, p} and s_{1} < s_{2} < ⋯ < s_{2q} in {1, 2, …, 2p}, such that

(i) The set {s_{1}, s_{2}, …, s_{2q}} is π̂-invariant and π̂(s_{i}) ≠ s_{i+1}, for all i in {1, 2, …, 2q − 1}.

(ii) If we put s_{0} = 0 and s_{2q+1} = 2p + 1, then for each i in {0, 1, …, 2q}, the set I_{i} = ]s_{i}, s_{i+1}[ ∩ ℤ is either the empty set or a non-crossing interval of integers for π̂.

Then {s_{1}, s_{2}, …, s_{2q}} = IRR(π̂).

Proof. It follows from (i), that there exists a (unique) permutation γ in S_{2q}, such that

π̂(s_{i}) = s_{γ(i)}, (i ∈ {1, 2, …, 2q}),

and moreover

γ(i) ≠ i + 1,  (i ∈ {1, 2, …, 2q − 1}).  (5.25)

Our first objective is to prove that γ is of the form ρ̂ for some (unique) permutation ρ in S_{q}^{irr}. For this, note first that by (ii), card(I_{i}) is an even number for all i in {0, 1, …, 2q}. Hence s_{i+1} − s_{i} is odd for all i in {0, 1, …, 2q}, and this implies that

s_{1}, s_{3}, …, s_{2q−1} are odd numbers
s_{2}, s_{4}, …, s_{2q} are even numbers

Since π̂^{2} = id and π̂(j) − j is odd for all j in {1, 2, …, 2p}, it follows now that γ^{2} = id and that γ(i) − i is odd for all i in {1, 2, …, 2q}. Therefore, by Remark 1.7(a), γ = ρ̂ for some (unique) ρ in S_{q}, and (5.25) shows that in fact ρ ∈ S_{q}^{irr}.

Returning now to the proof of the equation {s_{1}, s_{2}, …, s_{2q}} = IRR(π̂), note first that ∪_{i=0}^{2q}I_{i} ⊆ NC(π̂), and therefore

{s_{1}, s_{2}, …, s_{2q}} = {1, 2, …, 2p} \ ∪_{i=0}^{2q}I_{i} ⊇ IRR(π̂).

Suppose then that IRR(π̂) is a proper subset of {s_{1}, s_{2}, …, s_{2q}}. Then there exists j_{0} in {1, 2, …, 2q}, such that s_{j_{0}} ∈ NC(π̂), i.e., s_{j_{0}} ∈ I, for some non-crossing interval of integers for π̂. For this I, define

J = {j ∈ {1, 2, …, 2q} | s_{j} ∈ I}.

Then J ≠ ∅, and since s_{1} < s_{2} < ⋯ < s_{2q}, J is an interval of integers. Consider now the permutation ρ in S_{q}^{irr}, introduced above. Then, since π̂(I) = I, we have also that ρ̂(J) = J. Moreover, J is a non-crossing interval of integers for ρ̂. Indeed, if (a, b, c, d) were a crossing for ρ̂ contained in J, then clearly (s_{a}, s_{b}, s_{c}, s_{d}) would be a crossing for π̂ contained in I, which is impossible. Altogether, ρ is both irreducible and has a non-crossing interval of integers, and by Lemma 5.10(ii), this is impossible. Thus, we have reached a contradiction, which means that we must also have the inclusion {s_{1}, s_{2}, …, s_{2q}} ⊆ IRR(π̂). ■

5.13 Lemma. Let p be a positive integer, and let π be a permutation in S_{p} \ S_{p}^{nc}. Write then, as in Proposition 5.11, IRR(π̂) in the form

IRR(π̂) = {s_{1}, s_{2}, …, s_{2q}},

where q ∈ {1, …, p} and 1 ≤ s_{1} < s_{2} < ⋯ < s_{2q} ≤ 2p. Then s_{1}, s_{2}, …, s_{2q} satisfy, in addition, that

(i) s_{1}, s_{3}, …, s_{2q−1} are odd numbers.

(ii) s_{2}, s_{4}, …, s_{2q} are even numbers.

(iii) There is one and only one permutation ρ in S_{q}^{irr}, such that π̂(s_{j}) = s_{ρ̂(j)} for all j in {1, 2, …, 2q}.

Proof. This follows immediately from Proposition 5.11 and the first part of the proof of Proposition 5.12. ■

5.14 Definition. Let p be a positive integer, let π be a permutation in S_{p} \ S_{p}^{nc}, and let q, s_{1}, s_{2}, …, s_{2q} and I_{0}, I_{1}, …, I_{2q}, be as in Proposition 5.11. Then put

t_{i} = \frac{1}{2}card(I_{i}), (i ∈ \{0, 1, …, 2q\}),

and note that since I_{i} is either empty or a non-crossing interval of integers for π̂, t_{i} ∈ ℕ_{0} for all i. If t_{i} > 0, then as in Remark 5.6, we consider the order-preserving bijection φ_{i} of {1, 2, …, 2t_{i}} onto I_{i}, and we let π_{i} denote the (unique) permutation in S_{t_{i}}, satisfying that π̂_{i} = φ_{i}^{−1} ∘ (π̂_{|I_{i}}) ∘ φ. Clearly π_{i} ∈ S_{p}^{nc}.

It is convenient to consider the permutation group S_{0} of the empty set, as a group with one element π_{∅}. Then, in the setting considered above, we put π_{i} = π_{∅}, for all i in {0, 1, …, 2q}, for which t_{i} = 0. By convention, we put

k(π̂_{∅}) = 0,  and l(π̂_{∅}) = 1.  □  (5.26)

5.15 Lemma. Let p be a positive integer, let π be a permutation in S_{p} \ S_{p}^{nc}, and let ρ be the irreducible permutation introduced in Lemma 5.13(iii). Then σ(ρ̂) = σ(π̂).

Proof. Let q, s_{1}, s_{2}, …, s_{2q} and I_{0}, I_{1}, …, I_{2q}, be as in Proposition 5.11, and for each i in {0, 1, …, 2q}, let t_{i} and π_{i} be as in Definition 5.14. If t_{i} > 0, then π̂_{i} is non-crossing, and hence, by Proposition 1.23, π̂_{i} may be reduced to ê_{1} (where e_{1} is the permutation in S_{1}), by a series of successive cancellations of pairs. Here ê_{1} consists exactly of one pair of neighbours, so, formally speaking, ê_{1} can be reduced π̂_{∅}, by cancellation of this pair. Thus, π̂_{i} can be reduced to π̂_{∅}, by a series of successive cancellations of pairs, and forming the corresponding series of cancellations of pairs to π̂_{|I_{i}}, it follows that π̂ can be reduced to a permutation, which is, loosely speaking, obtained by “cutting out” π̂_{|I_{i}} from π̂. Forming these reductions for each i in {0, 1, …, 2q}, for which t_{i} > 0, it follows that π̂ can be reduced to ρ̂ by a series of successive cancellations of pairs. By Proposition 1.22, this implies that σ(π̂) = σ(ρ̂). ■

5.16 Proposition. Let p be a positive integer, let π be a permutation in S_{p} \ S_{p}^{nc}, and let q, s_{1}, s_{2}, …, s_{2q} be as in Proposition 5.11. Let further ρ be the permutation in S_{q}^{irr} introduced in Lemma 5.13(iii), and let π_{0}, π_{1}, …, π_{2q} be as in Definition 5.14. Then for any elements a_{1}, …, a_{r} of B(H, K) for which (5.2) holds, we have

∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} = c^{h(π̂)} ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}},  (5.27)

where

h(π̂) = k(π̂_{0}) + (l(π̂_{1}) − 1) + k(π̂_{2}) + ⋯ + (l(π̂_{2q−1}) − 1) + k(π̂_{2q}).  (5.28)

Proof. We start by introducing some notation. Let t be a positive integer, and let η be a permutation in S_{t}. We then put

Γ(η̂) = ∑_{1≤i_{1},... ,i_{t}≤r} a_{i_{1}}^{∗}a_{i_{η(1)}} ⋯ a_{i_{t}}^{∗}a_{i_{η(t)}},  (5.29)

and moreover, we put

Γ(π̂_{∅}) = 111_{B(H)}.  (5.30)

Note that Γ(η̂) can be expressed in terms of η̂ only, namely as

Γ(η̂) = ∑_{(i_{1},i_{2},i_{3},i_{4},... ,i_{2t})∈N(η̂)} a_{i_{1}}^{∗}a_{i_{2}}a_{i_{3}}^{∗}a_{i_{4}} ⋯ a_{i_{2t−1}}^{∗}a_{i_{2t}},  (5.31)

where

N(η̂) = {(i_{1}, i_{2}, …, i_{2t}) ∈ {1, 2, …, r}^{2t} | i_{j} = i_{η̂(j)}, for all j in {1, 2, …, 2t}}, (5.32)

(cf. Remark 1.7(b)). Consider next an interval of integers I, such that I ⊆ {1, 2, …, 2t} and η̂(I) = I. Write I in the form {α, α + 1, …, β}, and note that β − α + 1 = card(I) is an even number. We then put

N(η̂, I) = {(i_{α}, …, i_{β}) ∈ {1, 2, …, r}^{β−α+1} | i_{j} = i_{η̂(j)}, j = α, α + 1, …, β} (5.33)

and

Γ(η̂, I) = \begin{cases}∑_{(i_{α},... ,i_{β})∈N(η̂,I)}a_{i_{α}}^{∗}a_{i_{α+1}} ⋯ a_{i_{β−1}}^{∗}a_{i_{β}}, & if α is odd, \\ ∑_{(i_{α},... ,i_{β})∈N(η̂,I)}a_{i_{α}}a_{i_{α+1}}^{∗} ⋯ a_{i_{β−1}}a_{i_{β}}^{∗}, & if α is even. \end{cases}  (5.34)

Now, to prove (5.27), consider p in ℕ and π in S_{p} \ S_{p}^{nc}, and let q, s_{1}, s_{2}, …, s_{2q} and I_{0}, I_{1}, …, I_{2q}, t_{0}, t_{1}, …, t_{2q} be as in Proposition 5.11. Note then, that we may write N(π̂) as

N(π̂) = ⋃_{(i_{s_{1}},... ,i_{s_{2q}})∈N_{1}(π̂)} N(π̂, I_{0}) × {i_{s_{1}}} × N(π̂, I_{1}) × {i_{s_{2}}} × ⋯ × {i_{s_{2q}}} × N(π̂, I_{2p}), (5.35)

with the convention that N(π̂, I_{i}) is omitted in the product sets when 2t_{i} = card(I_{i}) = 0, and where

N_{1}(π̂) = {(i_{s_{1}}, …, i_{s_{2q}}) ∈ {1, 2, …, r}^{2q} | i_{s_{j}} = i_{π̂(s_{j})}, j = 1, 2, …, 2q}.  (5.36)

It follows thus, by (5.31), that

Γ(π̂) = ∑_{(i_{s_{1}},... ,i_{s_{2q}})∈N_{1}(π̂)} Γ(π̂, I_{0})a_{i_{s_{1}}}^{∗}Γ(π̂, I_{1})a_{i_{s_{2}}} ⋯ a_{i_{s_{2q}}} Γ(π̂, I_{2q}),  (5.37)

with the convention that if card(I_{i}) = 0,

Γ(π̂, I_{i}) = \begin{cases}111_{B(H)}, & if s_{i} is even, \\ 111_{B(K)}, & if s_{i} is odd. \end{cases}  (5.38)

To calculate Γ(π̂, I_{0}), …, Γ(π̂, I_{2q}), consider the non-crossing permutations π_{0}, π_{1}, …, π_{2q} introduced in Definition 5.14. Note then, that for each v in {0, 1, …, 2q}, such that t_{v} > 0, we have by a suitable relabeling of indices,

N(π̂, I_{v}) = {(i_{1}, i_{2}, …, i_{2t_{v}}) ∈ {1, 2, …, r}^{2t_{v}} | i_{j} = i_{π̂_{v}(j)}, j = 1, 2, …, 2t_{v}} = N(π̂_{v}).

It follows thus, that if t_{v} > 0,

Γ(π̂, I_{v}) = \begin{cases}∑_{1≤i_{1},... ,i_{tv} ≤r}a_{i_{1}}^{∗}a_{i_{πv(1)}} ⋯ a_{i_{tv}}^{∗} a_{i_{πv(tv)}}, & if v is even, \\ ∑_{1≤i_{1},... ,i_{tv} ≤r}a_{i_{1}}a_{i_{πv(1)}}^{∗} ⋯ a_{i_{tv}} a_{i_{πv(tv)}}^{∗}, & if v is odd, \end{cases}

and hence by Lemma 5.2 (since π̂_{v} is non-crossing),

Γ(π̂, I_{v}) = \begin{cases}c^{k(π̂_{v})}111_{B(H)}, & if v is even, \\ c^{l(π̂_{v})−1}111_{B(K)}, & if v is odd. \end{cases}  (5.39)

If t_{v} = 0, then by definition,

Γ(π̂, I_{v}) = \begin{cases}111_{B(H)}, & if v is even, \\ 111_{B(K)}, & if v is odd, \end{cases} = \begin{cases}c^{k(π̂_{v})}111_{B(H)}, & if v is even, \\ c^{l(π̂_{v})−1}111_{B(K)}, & if v is odd, \end{cases}  (5.40)

with k(π̂_{∅}), l(π̂_{∅}) as defined in (5.26). Combining (5.37),(5.39) and (5.40), it follows that with h(π̂) given in (5.28), we have

Γ(π̂) = c^{h(π̂)} ∑_{(i_{s_{1}},... ,i_{s_{2q}})∈N_{1}(π̂)} a_{i_{s_{1}}}^{∗}a_{i_{s_{2}}} ⋯ a_{i_{2q−1}}^{∗}a_{i_{s_{2q}}} .  (5.41)

Note finally, that with ρ the permutation introduced in Lemma 5.13(iii), we have that

N_{1}(π̂) = {(i_{1}, i_{2}, …, i_{2q}) ∈ {1, 2, …, r}^{2q} | i_{j} = i_{ρ̂(j)}, j = 1, 2, …, 2q} = N(ρ̂),

and therefore

∑_{(i_{s_{1}},... ,i_{s_{2q}})∈N_{1}(π̂)} a_{i_{s_{1}}}^{∗}a_{i_{s_{2}}} ⋯ a_{i_{2q−1}}^{∗}a_{i_{s_{2q}}} = ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}}.

Inserting this in (5.41), we obtain (5.27). ■

5.17 Definition. Let c be a positive number. Then for any p in ℕ_{0}, we define

g_{c}(p) = \begin{cases}\frac{1}{p} ∑_{j=1}^{p} \binom{p}{j}\binom{p}{j−1}c^{j}, & if p ∈ ℕ, \\ 1, & if p = 0, \end{cases}  (5.42)

and

h_{c}(p) = \begin{cases}\frac{1}{p} ∑_{j=1}^{p} \binom{p}{j}\binom{p}{j−1}c^{j−1}, & if p ∈ ℕ, \\ 1, & if p = 0. \end{cases}  (5.43)

Moreover, for p, q in ℕ_{0}, such that p ≥ q, we put

ν′(c, p, q) = ∑_{\begin{matrix}r_{0},r_{1},... ,r_{2q}≥0 \\ r_{0}+r_{1}+···+r_{2q}=p−q \end{matrix}} g_{c}(r_{0})h_{c}(r_{1})g_{c}(r_{2})h_{c}(r_{3}) ⋯ g_{c}(r_{2q}).  □  (5.44)

We are now ready to prove the main result of this section.

5.18 Theorem. Let a_{1}, …, a_{r} be elements of B(H, K), let c be a positive number, and assume that ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)}, and ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)}. Consider furthermore independent elements Y_{1}, …, Y_{r} of GRM(n, n, \frac{1}{n}), and put S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}. Then for any positive integer p,

\begin{matrix}𝔼[ & (S^{∗}S)^{p}] \\ = [ν′(c, p, 0)111_{B(H)} + ∑_{q=1}^{p} ν′(c, p, q)( ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)} ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}})] ⊗ 111_{n}. \end{matrix} (5.45)

Proof. Let p from ℕ be given. Then for each q in {1, 2, …, p}, we define

S_{p,q} = {π ∈ S_{p} | card(IRR(π̂)) = 2q},  (5.46)

and

M_{q} = ∑_{π∈S_{p,q}} n^{−2σ(π̂)}( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}})  (5.47)

It follows then by (5.1), that

𝔼[(S^{∗}S)^{p}] = [ ∑_{π∈S_{p}} n^{−2σ(π̂)}( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}})] ⊗ 111_{n} = ∑_{q=0}^{p} M_{q} ⊗ 111_{n}. (5.48)

By Lemma 5.10, S_{p,0} = S_{p}^{nc} and S_{p,p} = S_{p}^{irr}. Hence

M_{p} = ∑_{π∈S_{p}^{irr}} n^{−2σ(π̂)}( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}),  (5.49)

and by Corollary 5.4(i) and Corollary 1.24,

M_{0} = g_{c}(p)111_{B(H)} = ν′(c, p, 0)111_{B(H)}.  (5.50)

To calculate M_{1}, M_{2} …, M_{p−1}, we let, for each π in S_{p}, ρ(π) denote the irreducible permutation ρ associated to π in Lemma 5.13(iii). Then for any q in {1, 2, …, p − 1} and any ρ in S_{q}^{irr}, we define

R(p, ρ) = {π ∈ S_{p,q} | ρ(π) = ρ}.

Then we have the following disjoint union

S_{p,q} = ⋃_{ρ∈S_{q}^{irr}}^{•} R(p, ρ),

and therefore

M_{q} = ∑_{ρ∈S_{q}^{irr}} ∑_{π∈R(p,ρ)} n^{−2σ(π̂)}( ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}).  (5.51)

Note here, that for any ρ in S_{q}^{irr}, we have by Proposition 5.16 and Lemma 5.15,

\begin{aligned}∑_{π∈R(p,ρ)} n^{−2σ(π̂)}( & ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}}) \\ & = ( ∑_{π∈R(p,ρ)} c^{h(π̂)})n^{−2σ(ρ̂)}( ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}}), \end{aligned}  (5.52)

where for each π in R(p, ρ),

h(π̂) = k(π̂_{0}) + (l(π̂_{1}) − 1) + k(π̂_{2}) + ⋯ + (l(π̂_{2q−1}) − 1) + k(π̂_{2q}),

and where π_{0}, π_{1}, …, π_{2q} are the permutations introduced in Definition 5.14.

For any ρ in S_{q}^{irr} and any π in R(p, ρ), it follows from Proposition 5.11 and Lemma 5.13, that π̂ can be obtained from ρ̂ in a unique way, by “stuffing in” the intervals (or empty sets) I_{0}, I_{1}, …, I_{2q}, and the corresponding non-crossing permutations π̂_{0}, π̂_{1}, …, π̂_{2q}. Conversely, if π ∈ S_{p} such that π̂ can be obtained from ρ̂ by “stuffing in” intervals (or empty sets) J_{0}, J_{1}, …, J_{2q} and corresponding non-crossing permutations η̂_{0}, η̂_{1}, …, η̂_{2q}, then, by Proposition 5.12, π ∈ R(p, ρ) and J_{j} = I_{j}, η_{j} = π_{j}, for all j in {0, 1, …, 2q}. It follows thus, that the mapping

π ↦ (π_{0}, π_{1}, …, π_{2q})

is a bijection of R(p, ρ) onto the set of (2q + 1)-tuples (π_{0}, π_{1}, …, π_{2q}) of permutations for which there exist t_{0}, t_{1}, …, t_{2q} in ℕ_{0}, such that π_{i} ∈ S_{t_{i}}^{nc} for all i, and ∑_{i=0}^{2q} t_{i} = p − q (here we have used the convention that S_{0}^{nc} = S_{0} = {π_{∅}}).

Using this description of R(p, ρ), it follows that

∑_{π∈R(p,ρ)} c^{h(π̂)} = ∑_{\begin{matrix}t_{0},... ,t_{2q}≥0 \\ t_{0}+···+t_{2q}=p−q \end{matrix}} ∑_{π_{0}∈S_{t_{0}}^{nc},... ,π_{2q}∈S_{t_{2q}}^{nc}} c^{k(π̂_{0})}c^{(l(π̂_{1})−1)}c^{k(π̂_{2})} ⋯ c^{k(π̂_{2q})}.  (5.53)

Recall here from Definition 5.17 and Lemma 5.3, that for any t in ℕ,

∑_{η∈S_{t}^{nc}} c^{k(η̂)} = g_{c}(t),  and  ∑_{η∈S_{t}^{nc}} c^{l(η̂)−1} = h_{c}(t),

and by (5.26) this formula holds for t = 0 too. Using this in (5.53), it follows that

\begin{aligned}∑_{π∈R(p,ρ)} c^{h(π̂)} & = ∑_{\begin{matrix}t_{0},t_{1},... ,t_{2q}≥0 \\ t_{0}+t_{1}+···+t_{2q}=p−q \end{matrix}} g_{c}(t_{0})h_{c}(t_{1})g_{c}(t_{2})h_{c}(t_{3}) ⋯ g_{c}(t_{2q}) \\ & = ν′(c, p, q). \end{aligned}  (5.54)

Note, in particular, that the right hand side depends only on p and q, and not on ρ itself. Combining (5.51),(5.52) and (5.54), it follows that for any q in {1, 2, …, p − 1},

M_{q} = ν′(c, p, q) ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}( ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}}).  (5.55)

Since ν′(c, p, p) = 1, (5.55) holds for q = p too, by (5.49), and combining this with (5.48) and (5.50), we obtain, finally, (5.45). ■

5.19 Proposition. Let a_{1}, …, a_{r} in B(H, K), c in ]0, ∞[ and S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}, be as in Theorem 5.18. Then for any p in ℕ, we have that

\begin{aligned}∑_{π∈S_{p}} n^{−2σ(π̂)} & ∥ ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} ∥ \\ & = ν′(c, p, 0) + ∑_{q=1}^{p} ν′(c, p, q) ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}∥ ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}} ∥. \end{aligned}

Proof. This follows by exactly the same proof as for Theorem 5.18. ■

5.20 Example. Let a_{1}, …, a_{r} in B(H, K) and c from ]0, ∞[ be as in Theorem 5.18.

(a) For p = 1 or p = 2, we have S_{p} = S_{p}^{nc}. Hence by (5.1), Corollary 1.24 and Corollary 5.4(i), we get that

𝔼[S^{∗}S] = c111_{B(H)⊗M_{n}(ℂ)}, and 𝔼[(S^{∗}S)^{2}] = (c^{2} + c)111_{B(H)⊗M_{n}(ℂ)}.

This can also easily be obtained directly from (5.1) and (5.2).

(b) For p = 3, card(S_{3}) = 6 and card(S_{3}^{nc}) = 5. The only element of S_{3} \ S_{3}^{nc} is the irreducible permutation π given by

π(1) = 3, π(2) = 1, π(3) = 2.

For this π, σ(π̂) = 1, and it follows then by (5.1) and Corollary 5.4(i), that

𝔼[(S^{∗}S)^{3}] = (c^{3} + 3c^{2} + c)111_{B(H)⊗M_{n}(ℂ)} + (n^{−2} ∑_{i,j,k=1}^{r} a_{i}^{∗}a_{k}a_{j}^{∗}a_{i}a_{k}^{∗}a_{j}) ⊗ 111_{n}.

This follows also from Theorem 5.18, because S_{1}^{irr} = S_{2}^{irr} = ∅ and S_{3}^{irr} = {π}. □

6 The Sequence of Orthogonal Polynomials for the Measure μ_{c}

Throughout this section we consider a fixed positive constant c, and elements a_{1}, …, a_{r} of B(H, K), satisfying that

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(H)}.

Moreover, we put

S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i},

where Y_{1}, …, Y_{r} are independent elements of GRM(n, n, \frac{1}{n}).

As in Section 3, we let μ_{c} denote the probability measure on ℝ, given by

μ_{c} = max\{1 − c, 0\}δ_{0} + \frac{\sqrt{(x − a)(b − x)}}{2πx} · 1_{[a,b]}(x) · dx,

where a = (\sqrt{c} − 1)^{2}, b = (\sqrt{c} + 1)^{2}.

The asymptotic upper bound for the spectrum of S^{∗}S obtained in Section 4 (in the exact case), was obtained by making careful estimates of the moments 𝔼[(S^{∗}S)^{p}], p ∈ ℕ. However, these estimates cannot be used to give good asymptotic lower bounds for the spectrum of S^{∗}S in the case c > 1. To obtain such lower bounds, we shall instead consider the operators 𝔼[P_{q}^{c}(S^{∗}S)], where (P_{q}^{c})_{q∈ℕ_{0}} is the sequence of monic polynomials, obtained by Gram-Schmidt orthogonalization of the polynomials 1, x, x^{2}, …, w.r.t. the inner product

⟨f, g⟩ = ∫_{0}^{∞} f(x)\overline{g(x)} dμ_{c}(x), (f, g ∈ L^{2}(ℝ, μ_{c})).

The main result of this section is the equation

𝔼[P_{q}^{c}(S^{∗}S)] = [ ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}( ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}})] ⊗ 111_{n}, (q ∈ ℕ),

where S_{q}^{irr} is the set of permutations ρ in S_{q}, satisfying that

1 ≠ ρ(1) ≠ 2 ≠ ρ(2) ≠ ⋯ ≠ ρ(q)

(cf. Definition 1.16).

6.1 Proposition. Let (P_{q}^{c})_{q∈ℕ_{0}} be the sequence of polynomials on ℝ, defined by the recursion formulas:

P_{0}^{c}(x) = 1,  (6.1)
P_{1}^{c}(x) = x − c,  (6.2)
P_{q+1}^{c}(x) = (x − c − 1)P_{q}^{c}(x) − cP_{q−1}^{c}(x),  (q ≥ 1).  (6.3)

We then have

(i) For each q in ℕ_{0}, P_{q}^{c}(x) is a monic polynomial of degree q, and P_{q}^{c}(x) ∈ ℝ for all real numbers x.

(ii)  P_{q}^{c}(c + 1 + 2\sqrt{c} cos θ) = \frac{c^{\frac{q}{2}} sin((q + 1)θ) + c^{\frac{q−1}{2}} sin(qθ)}{sin θ}, (θ ∈ ]0, π[ ).

(iii)  ∫_{a}^{b} P_{q}^{c}(x)P_{q′}^{c}(x) dμ_{c}(x) = \begin{cases}c^{q}, & if q = q′, \\ 0, & if q ≠ q′, \end{cases}  (q, q′ ∈ ℕ_{0}).

In particular, (P_{q}^{c})_{q∈ℕ_{0}} is the sequence of monic orthogonal polynomials obtained by Gram-Schmidt orthogonalization of 1, x, x^{2}, …, in the Hilbert space L^{2}(ℝ, μ_{c}).

Proof. (i) This is clear from (6.1)-(6.3).

(ii) Consider the sequences (R_{q}^{c})_{q∈ℕ_{0}} and (T_{q}^{c})_{q∈ℕ_{0}} of polynomials, given by the recursion formulas

R_{0}^{c}(x) = 1,  (6.4)
R_{1}^{c}(x) = x − c − 1,  (6.5)
R_{q+1}^{c}(x) = (x − c − 1)R_{q}^{c}(x) − cR_{q−1}^{c}(x),  (q ≥ 1),  (6.6)

respectively

T_{0}^{c}(x) = 0,  (6.7)
T_{1}^{c}(x) = 1,  (6.8)
T_{q+1}^{c}(x) = (x − c − 1)T_{q}^{c}(x) − cT_{q−1}^{c}(x),  (q ≥ 1).  (6.9)

Note here, that the conditions (6.6) and (6.9) are the same, and therefore, the sequence (R_{q} + T_{q})_{q∈ℕ_{0}} satisfies this condition too. Moreover, the sequence (R_{q} + T_{q})_{q∈ℕ_{0}} also satisfies (6.1) and (6.2), and it follows thus, that

P_{q}^{c}(x) = R_{q}^{c}(x) + T_{q}^{c}(x), (q ∈ ℕ_{0}).

Note also, that T_{2}^{c}(x) = x − c − 1, so that the sequence (T_{q+1}^{c})_{q∈ℕ_{0}} satisfies (6.4)-(6.6), and hence

T_{q}^{c}(x) = R_{q−1}^{c}(x), (q ∈ ℕ).

Altogether, it follows that

P_{q}^{c}(x) = R_{q}^{c}(x) + R_{q−1}^{c}(x),  (q ≥ 1),  (6.10)
P_{0}^{c}(x) = R_{0}^{c}(x).  (6.11)

To prove (ii), it suffices therefore to show, that with x = c + 1 + 2\sqrt{c} cos θ, θ ∈ ]0, π[, one has

R_{q}^{c}(x) = \frac{c^{\frac{q}{2}} sin((q + 1)θ)}{sin θ},  (q ∈ ℕ_{0}).  (6.12)

For q = 0, this is clear from (6.4), and for q = 1, it follows easily from (6.5), using that sin 2θ = 2 sin θ cos θ. Proceeding by induction, assume now that p ≥ 1 and that (6.12) has been proved for all q in {0, 1, …, p}. Then by (6.6),

R_{p+1}^{c}(x) = \frac{2\sqrt{c} cos θ · c^{\frac{p}{2}} sin((p + 1)θ)}{sin θ} − \frac{c^{\frac{p+1}{2}} sin(pθ)}{sin θ},

when x = c + 1 + 2\sqrt{c} cos θ, θ ∈ ]0, π[. But 2 cos θ sin((p + 1)θ) = sin((p + 2)θ) + sin(pθ), and therefore

R_{p+1}^{c}(x) = \frac{c^{\frac{p+1}{2}} sin((p + 2)θ)}{sin θ},

which means that (6.12) holds for q = p + 1. Thus, by induction, (6.12) holds for all q in ℕ_{0}, and this concludes the proof of (ii).

(iii) We show first, that for any m, n in ℕ_{0},

∫_{0}^{∞} xR_{m}^{c}(x)R_{n}^{c}(x) dμ_{c}(x) = \begin{cases}0, & if n ≠ m, \\ c^{m+1}, & if n = m, \end{cases}  (6.13)

where R_{0}^{c}, R_{1}^{c}, R_{2}^{c}, …, are the polynomials determined by (6.4)-(6.6). Note for this, that if c < 1, then the atom for μ_{c} at 0, does not contribute to the integral on the left hand side of (6.13). Hence, for all values of c in ]0, ∞[, we have

∫_{0}^{∞} xR_{m}^{c}(x)R_{n}^{c}(x) dμ_{c}(x) = \frac{1}{2π}∫_{a}^{b}R_{m}^{c}(x)R_{n}^{c}(x)\sqrt{(x − a)(b − x)} dx.  (6.14)

By the substitution x = c + 1 + 2\sqrt{c} cos θ, θ ∈ ]0, π[, and by (6.12), the integral on the right hand side of (6.14) can be reduced to

\frac{2c}{π}∫_{0}^{π}c^{\frac{m+n}{2}} sin((m + 1)θ) sin((n + 1)θ) dθ,

which is equal to c^{m+1}δ_{m,n}. This proves (6.13).

We show next that

xR_{m}^{c}(x) = P_{m+1}^{c}(x) + cP_{m}^{c}(x),  (m ∈ ℕ_{0}).  (6.15)

For m = 0, this is clear from (6.1),(6.2) and (6.4), and for m ≥ 1, we get from (6.6) and (6.10), that

xR_{m}^{c}(x) = R_{m+1}^{c}(x) + (c + 1)R_{m}^{c}(x) + cR_{m−1}^{c}(x) = P_{m+1}^{c}(x) + cP_{m}^{c}(x).

This proves (6.15). Define now

γ_{m,n} = ∫_{0}^{∞} P_{m}^{c}(x)P_{n}^{c}(x) dμ_{c}(x), (m, n ∈ ℕ_{0}).

It follows then from (6.15), that

γ_{m+1,n} + cγ_{m,n} = ∫_{0}^{∞} xR_{m}^{c}(x)P_{n}^{c}(x) dμ_{c}(x), (m, n ∈ ℕ_{0}),

and applying then (6.10),(6.11) and (6.13), we get that

γ_{m+1,n} + cγ_{m,n} = c^{m+1}(δ_{m,n} + δ_{m,n−1}),  (m ∈ ℕ_{0}, n ∈ ℕ),  (6.16)

and

γ_{m+1,0} + cγ_{m,0} = c^{m+1}δ_{m,0},  (m ∈ ℕ_{0}).  (6.17)

Since μ_{c} is a probability measure, γ_{0,0} = 1, and using this and induction on (6.17), it follows that γ_{m,0} = 0 for all m in ℕ. Thus

γ_{0,n} = γ_{n,0} = \begin{cases}1, & if n = 0, \\ 0, & if n ≥ 1. \end{cases}  (6.18)

Consider now a fixed n in ℕ. By (6.16), we have then that

γ_{m+1,n} + cγ_{m,n} = \begin{cases}0, & if m ∈ \{0, 1, …, n − 2\}, \\ c^{n}, & if m = n − 1. \end{cases}

By induction in m (0 ≤ m ≤ n), we get then, by application of (6.18), that

γ_{m,n} = \begin{cases}0, & if m < n, \\ c^{n}, & if m = n, \end{cases}

and this completes the proof of (iii). ■

6.2 Lemma. For any non-negative integers p, q, put

ν(c, p, q) = c^{−q} ∫_{a}^{b} x^{p}P_{q}^{c}(x) dμ_{c}(x).  (6.19)

We then have

(i) For any p in ℕ_{0}, x^{p} = ∑_{q=0}^{p} ν(c, p, q)P_{q}^{c}(x).

(ii) For any p, q in ℕ_{0},

ν(c, p, q) ≥ 0, if q ≤ p,  (6.20)
ν(c, p, p) = 1,  (6.21)
ν(c, p, q) = 0, if q > p.  (6.22)

Proof. (i) Consider a fixed p from ℕ_{0}. By Proposition 6.1, span{P_{0}^{c}, P_{1}^{c}, …, P_{p}^{c}} is equal to the set of all polynomials of degree less than or equal to p. In particular we have that x^{p} = ∑_{q=0}^{p} γ_{q}P_{q}^{c}(x), for suitable complex numbers γ_{0}, …, γ_{p} (depending on c and p). Applying then the orthogonality relation in Proposition 6.1(iii), it follows that γ_{q} = ν(c, p, q) for all q in {0, 1, …, p}, and this proves (i).

(ii) By (6.1)-(6.3), it follows that

xP_{0}^{c}(x) = P_{1}^{c}(x) + cP_{0}^{c}(x),  (6.23)
xP_{q}^{c}(x) = P_{q+1}^{c}(x) + (c + 1)P_{q}^{c}(x) + cP_{q−1}^{c}(x),  (q ≥ 1),  (6.24)

so by induction in p, we get that x^{p}(= x^{p}P_{0}^{c}(x)), can be expressed as a linear combination of P_{0}^{c}(x), P_{1}^{c}(x), …, P_{p}^{c}(x), in which all coefficients are non-negative. By (i) and the linear independence of P_{0}^{c}(x), P_{1}^{c}(x), …, P_{p}^{c}(x), these coefficients are exactly ν(c, p, 0), ν(c, p, 1), …, ν(c, p, p), and hence (6.20) follows.

Note next that (6.21) follows from (i) and the facts that P_{p}^{c}(x) is a monic polynomial of degree p, whereas P_{0}^{c}(x), …, P_{p−1}^{c}(x) are all of degree at most p − 1.

Finally, (6.22) follows from (i) and the orthogonality relation in Proposition 6.1(iii). ■

6.3 Lemma. Let ν(c, p, q), p, q ∈ ℕ_{0}, be as in Lemma 6.2. Then for any fixed q in ℕ_{0}, the power series

∑_{p=0}^{∞} ν(c, p, q)t^{p},  (6.25)

converges for all t in the open complex ball B(0, \frac{1}{b}), where b = (\sqrt{c} + 1)^{2}. Moreover, the function

J_{q}^{c}(t) = ∑_{p=0}^{∞} ν(c, p, q)t^{p}, (t ∈ B(0, \frac{1}{b})),

is for all t in B(0, \frac{1}{b}) \ \{0\}, given by

J_{q}^{c}(t) = \frac{1 − (c − 1)t − \sqrt{(1 − at)(1 − bt)}}{2t} (\frac{1 − (c + 1)t − \sqrt{(1 − at)(1 − bt)}}{2ct})^{q} , (6.26)

where \sqrt{·} is the principal branch of the complex square-root.

Proof. Consider the Hilbert space L^{2}(ℝ, μ_{c}), and let A be the bounded operator on L^{2}(ℝ, μ_{c}), given by

[A(f)](x) = xf(x), (f ∈ L^{2}(ℝ, μ_{c}), x ∈ ℝ).

Note that A^{∗} = A and that sp(A) = supp(μ_{c}) ⊆ [0, b]. Thus, letting 111 denote the identity operator on L^{2}(ℝ, μ_{c}), 111 − tA is invertible for all complex numbers t such that |t| < \frac{1}{b}, and moreover, for such t,

(111 − tA)^{−1} = ∑_{p=0}^{∞} t^{p}A^{p}, (norm convergence).

For any t in B(0, \frac{1}{b}), we have thus that

∑_{p=0}^{∞} ν(c, p, q)t^{p} = c^{−q} ∑_{p=0}^{∞} ⟨x^{p}, P_{q}^{c}⟩t^{p} = c^{−q} ∑_{p=0}^{∞} ⟨A^{p}P_{0}^{c}, P_{q}^{c}⟩t^{p} = c^{−q}⟨(1 − tA)^{−1}P_{0}^{c}, P_{q}^{c}⟩.

This shows that the series in (6.25) converges for all t in B(0, \frac{1}{b}), and morever, that

J_{q}^{c}(t) = c^{−q}⟨(1 − tA)^{−1}P_{0}^{c}, P_{q}^{c}⟩,  (t ∈ B(0, \frac{1}{b})).  (6.27)

To prove (6.26), we shall calculate the right hand side of (6.27). For this, consider for each z in B(0, \frac{1}{\sqrt{c}}) the series ∑_{q=0}^{∞} z^{q}P_{q}^{c}, and note that by Lemma 6.1(iii), this series converges in ∥ · ∥_{2}-norm in L^{2}(ℝ, μ_{c}). We may thus define

ω_{z} = ∑_{q=0}^{∞} z^{q}P_{q}^{c} ∈ L^{2}(ℝ, μ_{c}),  (z ∈ B(0, \frac{1}{\sqrt{c}})).  (6.28)

With A as above, it follows now by (6.23) and (6.24), that for any z in B(0, \frac{1}{\sqrt{c}}) \ \{0\},

Aω_{z} = ∑_{n=0}^{∞} z^{n}AP_{n}^{c} = cP_{0}^{c} + P_{1}^{c} + ∑_{n=1}^{∞} z^{n}(cP_{n−1}^{c} + (c + 1)P_{n}^{c} + P_{n+1}^{c})

= (c + cz)P_{0}^{c} + ∑_{n=1}^{∞}(z^{n−1} + (c + 1)z^{n} + cz^{n+1})P_{n}^{c}

= (c + cz)P_{0}^{c} + z^{−1}(1 + (c + 1)z + cz^{2}) ∑_{n=1}^{∞} z^{n}P_{n}^{c}

= (c + cz − z^{−1}(1 + (c + 1)z + cz^{2}))P_{0}^{c} + z^{−1}(1 + (c + 1)z + cz^{2})ω_{z} = −z^{−1}(1 + z)P_{0}^{c} + z^{−1}(1 + z)(1 + cz)ω_{z},

where the infinite sums converge in ∥ · ∥_{2}-norm. From this it follows that

(z^{−1}(1 + z)(1 + cz)111 − A)ω_{z} = z^{−1}(1 + z)P_{0}^{c}, (z ∈ B(0, \frac{1}{\sqrt{c}}) \ \{0\}),

and hence that

(111 − \frac{z}{(1+z)(1+cz)}A)ω_{z} = \frac{1}{1+cz}P_{0}^{c},  (z ∈ B(0, \frac{1}{\sqrt{c}}) \ \{−1, −\frac{1}{c}\}).  (6.29)

Define now

φ(z) = \frac{z}{(1 + z)(1 + cz)}, (z ∈ ℂ \ \{−1, \frac{1}{c}\}).

Since sp(A) ⊆ [0, b], it follows that (111 − φ(z)A) is invertible whenever φ(z) ∉ [\frac{1}{b}, ∞[, and in particular, as long as |φ(z)| < \frac{1}{b}. Note then, that φ is analytic on ℂ \ \{−1, −\frac{1}{c}\}, and that φ(0) = 0, φ′(0) = 1. It follows thus, that we may choose neighbourhoods U and V of 0 in ℂ, such that φ is a bijection of U onto V. We may assume, in addition, that

U ⊆ B(0, \frac{1}{\sqrt{c}}) \ \{−1, −\frac{1}{c}\}, and V ⊆ B(0, \frac{1}{b}).

For z in U, it follows now from (6.29), that

ω_{z} = \frac{1}{1+cz}(1 − φ(z)A)^{−1}P_{0}^{c},

and hence, by (6.27) and Lemma 6.1(iii),

J_{q}^{c}(φ(z)) = (1 + cz) · c^{−q}⟨ω_{z}, P_{q}^{c}⟩ = (1 + cz)z^{q},  (z ∈ U).  (6.30)

It remains to invert φ. By solving the equation

t = \frac{z}{(1 + z)(1 + cz)},

w.r.t. z, we find that

φ^{−1}(t) = \frac{1 − (c + 1)t ± \sqrt{(1 − at)(1 − bt)}}{2ct}, (t ∈ V \ \{0\}),

where, as usual, a = (\sqrt{c} − 1)^{2} and b = (\sqrt{c} + 1)^{2}. Since φ^{−1}(t) → 0 as t → 0, it follows that for some neighbourhood V_{0} of 0, such that V_{0} ⊆ V, we must have

φ^{−1}(t) = \frac{1 − (c + 1)t − \sqrt{(1 − at)(1 − bt)}}{2ct},  (t ∈ V_{0} \ \{0\}),  (6.31)

where \sqrt{·} is the principal part of the square root. Hence, we have also that

1 + cφ^{−1}(t) = \frac{1 − (c − 1)t − \sqrt{(1 − at)(1 − bt)}}{2t},  (t ∈ V_{0} \ \{0\}).  (6.32)

Inserting (6.31) and (6.32) in (6.30), we obtain that (6.26) holds for all t in V_{0} \ {0}.

To show that (6.26) actually holds for all t in B(0, \frac{1}{b}) \ \{0\}, note that for all such t, Re(1 − at) > 0 and Re(1 − bt) > 0, so that (1 − at)(1 − bt) ∈ ℂ \] − ∞, 0]. Hence, with \sqrt{·} the principal branch of the square root, t ↦ \sqrt{(1 − at)(1 − bt)} is an analytic function of t ∈ B(0, \frac{1}{b}). By uniqueness of analytic continuation, it follows thus, that (6.26) holds for all t in B(0, \frac{1}{b}) \ \{0\}. ■

6.4 Lemma. Let g_{c}(p) and h_{c}(p), p ∈ ℕ_{0}, be as in Definition 5.17. Then the power series

G_{c}(t) = ∑_{p=0}^{∞} g_{c}(p)t^{p},  (6.33)

and

H_{c}(t) = ∑_{p=0}^{∞} h_{c}(p)t^{p},  (6.34)

are convergent for all t in B(0, \frac{1}{b}), and

J_{q}^{c}(t) = t^{q}G_{c}(t)^{q+1}H_{c}(t)^{q},  (t ∈ B(0, \frac{1}{b})).  (6.35)

Proof. By (5.10), we have

g_{c}(p) = ∫_{0}^{∞} x^{p} dμ_{c}(x), (p ∈ ℕ),

and since g_{c}(0) = 1, the same formula holds for p = 0. Hence g_{c}(p) = ν(c, p, 0), for all p in ℕ_{0}, so by Lemma 6.3, the series in (6.33) converges for all t in B(0, \frac{1}{b}), and

G_{c}(t) = J_{0}^{c}(t) = \frac{1 − (c − 1)t − \sqrt{(1 − at)(1 − bt)}}{2t},  (t ∈ B(0, \frac{1}{b}) \ \{0\}).  (6.36)

Since h_{c}(0) = 1 and since h_{c}(p) = \frac{1}{c}g_{c}(p), for all p in ℕ, the series in (6.34) is also convergent for all t in B(0, \frac{1}{b}), and

H_{c}(t) = 1 + \frac{1}{c}(G_{c}(t) − 1), (t ∈ B(0, \frac{1}{c})).

Hence by (6.34)

H_{c}(t) = \frac{1 + (c − 1)t − \sqrt{(1 − at)(1 − bt)}}{2ct},  (t ∈ B(0, \frac{1}{b}) \ \{0\}).  (6.37)

By (6.36) and (6.37), we get now for all t in B(0, \frac{1}{b}) \ \{0\},

G_{c}(t)H_{c}(t) = \frac{(1 − \sqrt{(1 − at)(1 − bt)})^{2} − (c − 1)^{2}t^{2}}{4ct^{2}}

= \frac{1 + (1 − at)(1 − bt) − 2\sqrt{(1 − at)(1 − bt)} − (c − 1)^{2}t^{2}}{4ct^{2}}

= \frac{1 + (1 − 2(c + 1)t + (c − 1)^{2}t^{2}) − 2\sqrt{(1 − at)(1 − bt)} − (c − 1)^{2}t^{2}}{4ct^{2}}

= \frac{1 − (c + 1)t − \sqrt{(1 − at)(1 − bt)}}{2ct^{2}}.

Combining this with (6.36) and (6.26), it follows that

J_{c}^{q}(t) = G_{c}(t)(tG_{c}(t)H_{c}(t))^{q}, (t ∈ B(0, \frac{1}{b})),

and the same formula holds trivially for t = 0, by (6.22). This proves (6.35). ■

6.5 Lemma. For all p, q in ℕ_{0} such that p ≥ q, let ν(c, p, q) be as introduced in Definition 5.17. Then

ν′(c, p, q) = ν(c, p, q), (p, q ∈ ℕ_{0}, q ≤ p).

Proof. Recall from Definition 5.17, that for p, q in ℕ_{0}, such that p ≥ q, we have

ν′(c, p, q) = ∑_{\begin{matrix}r_{0},r_{1},... ,r_{2q}≥0 \\ r_{0}+r_{1}+···+r_{2q}=p−q \end{matrix}} g_{c}(r_{0})h_{c}(r_{1})g_{c}(r_{2})h_{c}(r_{3}) ⋯ g_{c}(r_{2q}).

Hence ν′(c, p, q) is the coefficient to t^{p−q} in the power series for

G_{c}(t)H_{c}(t)G_{c}(t)H_{c}(t) ⋯ G_{c}(t), (2q + 1 factors),

and therefore ν′(c, p, q) is the coefficient to t^{p} in the power series for t^{q}G_{c}(t)^{q+1}H_{c}(t)^{q}. Thus, by Lemma 6.3 and Lemma 6.4, it follows that

ν′(c, p, q) = ν(c, p, q), for all p, q in ℕ_{0}, such that p ≥ q. ■

6.6 Theorem. Let H, K be Hilbert spaces, and let a_{1}, …, a_{r} be elements of B(H, K), satisfying that ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)}, for some positive real number c. Furthermore, let Y_{1}, …, Y_{r} be independent elements of GRM(n, n, \frac{1}{n}), and put S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}. Then for any q in ℕ,

𝔼[P_{q}^{c}(S^{∗}S)] = [ ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}( ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}})] ⊗ 111_{n}.

Proof. For each q in ℕ, put

T_{q} = ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}( ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}}),

and put T_{0} = 111_{B(H)}. By Theorem 5.18 and Lemma 6.5, it follows then that

𝔼[(S^{∗}S)^{p}] = ∑_{q=0}^{p} ν(c, p, q) · T_{q} ⊗ 111_{n},  (p ∈ ℕ_{0}).  (6.38)

On the other hand, it follows from Lemma 6.2(i), that

𝔼[(S^{∗}S)^{p}] = ∑_{q=0}^{p} ν(c, p, q)𝔼[P_{q}^{c}(S^{∗}S)],  (p ∈ ℕ_{0}).  (6.39)

We prove that

𝔼[P_{q}^{c}(S^{∗}S)] = T_{q} ⊗ 111_{n},  (q ∈ ℕ_{0}),  (6.40)

by induction in q. Note that (6.40) is trivial for q = 0. Consider then p from ℕ, and assume that (6.40) has been proved for q = 0, 1, …, p − 1. Since ν(c, p, p) = 1, by Lemma 6.2(ii), it follows then from (6.39) and (6.38), that

𝔼[P_{p}^{c}(S^{∗}S)] = 𝔼[(S^{∗}S)^{p}] − ∑_{q=0}^{p−1} ν(c, p, q)𝔼[P_{q}^{c}(S^{∗}S)]

= 𝔼[(S^{∗}S)^{p}] − ∑_{q=0}^{p−1} ν(c, p, q) · T_{q} ⊗ 111_{n}

= T_{p} ⊗ 111_{n}.

Thus, (6.40) holds for q = p, and this completes the proof. ■

6.7 Example. By (6.1)-(6.3), it follows that

P_{1}^{c}(x) = x − c,  (6.41)
P_{2}^{c}(x) = x^{2} − (2c + 1)x + c^{2},  (6.42)
P_{3}^{c}(x) = x^{3} − (3c + 2)x^{2} + (3c^{2} + 2c + 1)x − c^{3}.  (6.43)

By Example 5.20, S_{p}^{irr} = ∅ if p ∈ {1, 2}, and S_{3}^{irr} = {π}, where π is the permutation given by π(1) = 3, π(2) = 1, π(3) = 2, so that σ(π̂) = 1. It follows thus by Theorem 6.6, that

𝔼[P_{1}^{c}(S^{∗}S)] = 0,
𝔼[P_{2}^{c}(S^{∗}S)] = 0,
𝔼[P_{3}^{c}(S^{∗}S)] = n^{−2} ∑_{i,j,k=1}^{r} a_{i}^{∗}a_{k}a_{j}^{∗}a_{i}a_{k}^{∗}a_{j}.

These three formulas can also easily be derived directly from Example 5.20, using the formulas (6.41)-(6.43). □

7 An Upper Bound for 𝔼[ exp(−tS^{∗}S)], t ≥ 0

Throughout this section, we consider elements a_{1}, …, a_{r} of B(H, K) (for given Hilbert spaces H and K), satisfying that

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)},

for some constant c in [1, ∞[. Moreover, we consider independent elements Y_{1}, …, Y_{r} of GRM(n, n, \frac{1}{n}), and put

S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}.

As in Section 3, we let μ_{c} denote the probability measure on ℝ, given by

μ_{c} = \frac{\sqrt{(x − a)(b − x)}}{2πx} · 1_{[a,b]}(x) · dx,

where a = (\sqrt{c} − 1)^{2} and b = (\sqrt{c} + 1)^{2}. Furthermore, we let (P_{q}^{c})_{q∈ℕ_{0}} be the sequence of monic orthogonal polynomials w.r.t. μ_{c} as defined in Section 6. In particular P_{0}^{c} ≡ 1.

7.1 Lemma. Let, as above, a = (\sqrt{c} − 1)^{2} and b = (\sqrt{c} + 1)^{2}. Then for any q in ℕ_{0},

(i) P_{q}^{c}(x) ≥ P_{q}^{c}(b) > 0, for all x in ]b, ∞[.

(ii) |P_{q}^{c}(x)| ≤ P_{q}^{c}(b), for all x in [a, b].

(iii) |P_{q}^{c}(x)| ≤ P_{q}(2c + 2 − x), for all x in ] − ∞, a[.

Proof. We start by proving (ii). If x ∈ [a, b], then x = c + 1 + 2\sqrt{c} cos θ, for some θ in [0, π]. For θ in ]0, π[, we have from Proposition 6.1(ii), that

P_{q}^{c}(c + 1 + 2\sqrt{c} cos θ) = \frac{c^{\frac{q}{2}} sin((q + 1)θ) + c^{\frac{q−1}{2}} sin(qθ)}{sin θ}.  (7.1)

Note here that for any k in ℕ_{0},

\frac{sin((k + 1)θ)}{sin θ} = e^{−kθ}(1 + e^{2iθ} + e^{4iθ} + ⋯ + e^{2kiθ}),  (7.2)

so that |\frac{sin((k+1)θ)}{sin θ} | ≤ k + 1. It follows thus that

|P_{q}^{c}(x)| ≤ c^{\frac{q}{2}} (q + 1) + c^{\frac{q−1}{2}} q,  (x ∈]a, b[),  (7.3)

and by continuity, (7.3) holds also for x = a and x = b. By (7.2), lim_{θ→0} \frac{sin((k+1)θ)}{sin θ} = k +1, for any k in ℕ_{0}, and hence the right hand side of (7.3) is equal to P_{q}^{c}(b). This proves (ii).

To prove (i), we note first, that by uniqueness of analytic continuation, (7.1) actually holds for all θ in ℂ \ πℤ. If we put θ = iρ, ρ > 0, we get the equation:

P_{q}^{c}(c + 1 + 2\sqrt{c} cosh ρ) = \frac{c^{\frac{q}{2}} sinh((q + 1)ρ) + c^{\frac{q−1}{2}} sinh(qρ)}{sinh ρ},  (ρ ∈]0, ∞[), (7.4)

which covers the values of P_{q}(x) for all x in ]b, ∞[. Note here that for any k in ℕ_{0},

\frac{sinh((k + 1)ρ)}{sinh ρ} = e^{−kρ}(1 + e^{2ρ} + e^{4ρ} + ⋯ + e^{2kρ}),

and hence, if k is even,

\frac{sinh((k + 1)ρ)}{sinh ρ} = 1 + 2 cosh(2ρ) + 2 cosh(4ρ) + ⋯ + 2 cosh(kρ),

whereas, if k is odd,

\frac{sinh((k + 1)ρ)}{sinh ρ} = 2 cosh(ρ) + 2 cosh(3ρ) + ⋯ + 2 cosh(kρ),

so in both cases \frac{sinh((k+1)ρ)}{sin ρ} is an increasing function of ρ > 0. It follows thus from (7.4), that P_{q}^{c}(x) ≥ P_{q}^{c}(b) for all x in ]b, ∞[. Moreover, as we saw in the proof of (ii), P_{q}^{c}(b) > 0. This concludes the proof of (i).

Finally, to prove (iii), we put θ = π + iρ in (7.1), and get for ρ in ]0, ∞[, that

|P_{q}^{c}(c + 1 − 2\sqrt{c} cosh ρ)| = |\frac{(−1)^{q}c^{\frac{q}{2}} sinh((q + 1)ρ) + (−1)^{q−1}c^{\frac{q−1}{2}} sinh(qρ)}{sinh ρ}|

≤ \frac{c^{\frac{q}{2}} sinh((q + 1)ρ) + c^{\frac{q−1}{2}} sinh(qρ)}{sinh ρ}

= P_{q}^{c}(c + 1 + 2\sqrt{c} cosh ρ).

This proves (iii). ■

7.2 Definition. For each q in ℕ_{0}, we define the function ψ_{q}^{c}: ℝ → ℝ, by the equation

ψ_{q}^{c}(t) = c^{−q} ∫_{a}^{b} exp(tx)P_{q}^{c}(x) dμ_{c}(x),  (t ∈ ℝ).  □

7.3 Lemma. Consider the sequence (ψ_{q}^{c})_{q∈ℕ_{0}} of functions, introduced in Definition 7.2, and for each p in ℕ_{0}, let, as in Section 6,

ν(c, p, q) = c^{−q} ∫_{a}^{b} x^{p}P_{q}^{c}(x) dμ_{c}(x), (p, q ∈ ℕ_{0}).

We then have

(i) ψ_{q}^{c}(t) = ∑_{p=q}^{∞} \frac{t^{p}}{p!} ν(c, p, q), for all t in ℝ.

(ii) ∑_{q=0}^{∞} |ψ_{q}^{c}(t)| · |P_{q}^{c}(x)| ≤ exp(|t|x) + exp(|t|(2c + 2)), for all t in ℝ and all x in [0, ∞[.

(iii) exp(tx) = ∑_{q=0}^{∞} ψ_{q}^{c}(t) · P_{q}^{c}(x), for all t in ℝ and x in [0, ∞[, and for fixed t in ℝ, the series converges uniformly in x on compact subsets of [0, ∞[.

Proof. (i) By Lemma 6.2(ii), ν(c, p, q) = 0 whenever q > p. Hence (i) follows from the power series expansion of exp(tx).

(ii) Let β: ℝ → [b, ∞[ be the continuous function defined by:

β(x) = \begin{cases}x, & if x > b, \\ b, & if a ≤ x ≤ b, \\ 2c + 2 − x, & if x < a, \end{cases}

It follows then from Lemma 7.1, that

|P_{q}^{c}(x)| ≤ P_{q}^{c}(β(x)),  (x ∈ ℝ, q ∈ ℕ_{0}).  (7.5)

Recall that x^{p} = ∑_{q=0}^{p} ν(c, p, q)P_{q}^{c}(x), for all p in ℕ (c.f. Lemma 6.2(i)). Hence, for x, t in ℝ, we have that

exp(tx) = ∑_{p=0}^{∞} \frac{t^{p}}{p!} x^{p} = ∑_{p=0}^{∞} \frac{t^{p}}{p!} (∑_{q=0}^{p}ν(c, p, q)P_{q}^{c}(x)).  (7.6)

Substituting x with β(x) and t with |t| in this formula, and recalling from Lemma 6.2(ii), that ν(c, p, q) ≥ 0, for 0 ≤ q ≤ p, we get by application of (7.5),

∑_{p=0}^{∞} \frac{|t|^{p}}{p!} (∑_{q=0}^{p}ν(c, p, q)|P_{q}^{c}(x)|) ≤ ∑_{p=0}^{∞} \frac{|t|^{p}}{p!} (∑_{q=0}^{p}ν(c, p, q)P_{q}^{c}(β(x))) = exp(|t|β(x)) < ∞.

Hence, we can apply Fubini’s theorem to the double sum in (7.6), and obtain that

exp(tx) = ∑_{q=0}^{∞} ( ∑_{p=q}^{∞} \frac{t^{p}}{p!} ν(c, p, q))P_{q}^{c}(x),  (x, t ∈ ℝ).  (7.7)

Similarly we have that

exp(|t|β(x)) = ∑_{q=0}^{∞} ( ∑_{p=q}^{∞} \frac{|t|^{p}}{p!} ν(c, p, q))P_{q}^{c}(β(x)),  (x, t ∈ ℝ).  (7.8)

Note here that by (i) proved above, we have that,

|ψ_{q}^{c}(t)| ≤ ∑_{p=q}^{∞} \frac{|t|^{p}}{p!} ν(c, p, q).  (7.9)

Since β(x) ≤ max{2c + 2, x} for all x in [0, ∞[, (7.5) and (7.7)-(7.9) imply that for all t in ℝ and x in [0, ∞[,

∑_{q=0}^{∞} |ψ_{q}^{c}(t)| · |P_{q}^{c}(x)| ≤ exp(|t|β(x)) ≤ exp(|t|(2c + 2)) + exp(|t|x),

and this proves (ii).

(iii) The summation formula in (iii) follows from (i) and (7.7). To prove that the convergence is uniform in x on compact subsets, we observe that for any Q in ℕ,

\begin{aligned}| exp(tx) − ∑_{q=0}^{Q} ψ_{q}^{c}(t)P_{q}^{c}(x)| & ≤ ∑_{q=Q+1}^{∞} |ψ_{q}^{c}(t)| · |P_{q}^{c}(x)| ≤ ∑_{q=Q+1}^{∞} ( ∑_{p=q}^{∞} \frac{|t|^{p}}{p!} ν(c, p, q)P_{q}^{c}(β(x))) \\ & ≤ ∑_{p=Q+1}^{∞} \frac{|t|^{p}}{p!} (∑_{q=0}^{p}ν(c, p, q)P_{q}^{c}(β(x))) = ∑_{p=Q+1}^{∞} \frac{(|t|β(x))^{p}}{p!} . \end{aligned} (7.10)

Since β is continuous, and hence bounded on compact sets, it follows readily from (7.10) that for fixed t in ℝ, the series in (iii) converges uniformly in x on compact subsets of [0, ∞[. ■

7.4 Proposition. Consider the sequence (ψ_{q}^{c})_{q∈ℕ_{0}} of functions, introduced in Definition 7.2. Then for any t in ℝ such that |t| < \frac{n}{c}, the function ω ↦ exp(tS^{∗}(ω)S(ω)) is integrable in the sense of Definition 3.1, and

𝔼[ exp(tS^{∗}S)] = ∑_{q=0}^{∞} ψ_{q}^{c}(t)𝔼[P_{q}^{c}(S^{∗}S)],  (7.11)

where the sum on the right hand side is absolutely convergent in B(H^{n}).

Proof. We start by proving that the right hand side of (7.11) is absolutely convergent in B(H^{n}). Since |ψ_{q}^{c}(t)| ≤ ψ_{q}^{c}(|t|) by Lemma 7.3(i) and (7.9), it suffices to consider the case where t ≥ 0.

By Lemma 7.3(i), we have for any t in [0, ∞[,

∑_{q=0}^{∞} ψ_{q}^{c}(t)∥𝔼[P_{q}^{c}(S^{∗}S)]∥ = ∑_{p=0}^{∞} \frac{t^{p}}{p!}( ∑_{q=0}^{p} ν(c, p, q)∥𝔼[P_{q}^{c}(S^{∗}S)]∥).  (7.12)

Note here, that by Theorem 6.6,

∥𝔼[P_{q}^{c}(S^{∗}S)]∥ ≤ ∑_{ρ∈S_{q}^{irr}} n^{−2σ(ρ̂)}∥ ∑_{1≤i_{1},... ,i_{q}≤r} a_{i_{1}}^{∗}a_{i_{ρ(1)}} ⋯ a_{i_{q}}^{∗}a_{i_{ρ(q)}} ∥,

for any q in ℕ, whereas

∥𝔼[P_{0}^{c}(S^{∗}S)]∥ = ∥𝔼(111_{B(H^{n})})∥ = 1.

Hence, by Proposition 5.19, Lemma 6.5 and Proposition 2.7, we have for any p in ℕ,

\begin{aligned}∑_{q=0}^{p} ν(c, p, q)∥𝔼[P_{q}^{c}(S^{∗}S)]∥ & ≤ ∑_{π∈S_{p}} n^{−2σ(π̂)}∥ ∑_{1≤i_{1},... ,i_{p}≤r} a_{i_{1}}^{∗}a_{i_{π(1)}} ⋯ a_{i_{p}}^{∗}a_{i_{π(p)}} ∥ \\ & ≤ ∑_{π∈S_{p}} n^{−2σ(π̂)}c^{κ(π̂)}. \end{aligned}  (7.13)

Using now that c ≥ 1, and that κ(π̂) ≤ k(π̂) + 2σ(π̂) (c.f. Proposition 2.10), it follows that for any p in ℕ,

∑_{π∈S_{p}} n^{−2σ(π̂)}c^{κ(π̂)} ≤ ∑_{π∈S_{p}} (\frac{n}{c})^{−2σ(π̂)}c^{k(π̂)}.  (7.14)

For p = 0, we note that

∑_{q=0}^{p} ν(c, p, q)∥𝔼[P_{q}^{c}(S^{∗}S)]∥ = 1.  (7.15)

Combining now (7.12)-(7.15), we get that

∑_{q=0}^{∞} ψ_{q}^{c}(t)∥𝔼[P_{q}^{c}(S^{∗}S)]∥ ≤ 1 + ∑_{p=1}^{∞} \frac{t^{p}}{p!} (∑_{π∈S_{p}}(\frac{n}{c})^{−2σ(π̂)}c^{k(π̂)}).  (7.16)

Using then that −2σ(π̂) = k(π̂) + l(π̂) − p − 1, it follows that

\begin{aligned}∑_{q=0}^{∞} ψ_{q}^{c}(t)∥𝔼[P_{q}^{c}(S^{∗}S)]∥ & ≤ 1 + ∑_{p=1}^{∞} \frac{1}{p!}(\frac{ct}{n})^{p} ∑_{π∈S_{p}}n^{k(π̂)}(\frac{n}{c})^{l(π̂)−1} \\ & ≤ 1 + ct ∑_{p=1}^{∞} \frac{1}{(p−1)!}(\frac{ct}{n})^{p−1} ∑_{π∈S_{p}}n^{k(π̂)−1}(\frac{n}{c})^{l(π̂)−1}, \end{aligned}  (7.17)

where the last equality follows by noting that \frac{1}{p!} ≤ \frac{1}{(p−1)!} for all p in ℕ. By Lemma 3.4, the last quantity in (7.17) is finite whenever 0 ≤ \frac{ct}{n} < 1, and this shows that the right hand side of (7.11) is absolutely convergent for all t in ] − \frac{n}{c}, \frac{n}{c}[, as desired.

It remains now (cf. Definition 3.1) to show, that for any state φ on B(H^{n}),

𝔼[φ(exp(tS^{∗}S))] = ∑_{q=0}^{∞} ψ_{q}^{c}(t)φ(𝔼[P_{q}^{c}(S^{∗}S)]),  (t ∈]−\frac{n}{c}, \frac{n}{c}[).  (7.18)

So consider a fixed t from ] − \frac{n}{c}, \frac{n}{c}[ and a fixed state φ on B(H^{n}). Since the spectrum of S^{∗}(ω)S(ω) is compact for each ω in Ω, it follows then by Lemma 7.3, that

φ[ exp(tS^{∗}(ω)S(ω))] = ∑_{p=0}^{∞} ψ_{q}^{c}(t)φ[P_{q}^{c}(S_{n}(ω)^{∗}S_{n}(ω))],  (7.19)

so we need to show that we can integrate termwise in the sum on the right hand side. Note for this, that by Lemma 7.3(ii), and the function calculus for selfadjoint operators on Hilbert spaces,

∑_{p=0}^{∞} |ψ_{q}^{c}(t)| · |P_{q}^{c}(S(ω)^{∗}S(ω))| ≤ exp(2(c + 1)|t|)111_{B(H^{n})} + exp(|t|S(ω)^{∗}S(ω)), (7.20)

where |T | = (T^{2})^{\frac{1}{2}} , for any selfadjoint T in B(H^{n}). For such T, we have also that |φ(T)| ≤ φ(|T |), and hence it follows from (7.20), that

∑_{p=0}^{∞} |ψ_{q}^{c}(t)| · |φ[P_{q}^{c}(S(ω)^{∗}S(ω))]| ≤ exp(2(c + 1)|t|) + φ[ exp(|t|S(ω)^{∗}S(ω))]. (7.21)

Since 𝔼[φ(exp(|t|S^{∗}S))] < ∞, by Proposition 3.2, it follows from (7.21) and Lebesgue’s Theorem on Dominated Convergence, that we may integrate termwise in (7.19), and hence obtain (7.18). This concludes the proof. ■

In order to obtain the upper bound for 𝔼[ exp(−tS^{∗}S)] in Theorem 7.8 below, we need more precise information about the behaviour of the function ψ_{q}^{c}(t) for t < 0.

7.5 Proposition. Consider the sequence (ψ_{q}^{c})_{q∈ℕ_{0}} of functions, defined in Definition 7.2. Then for any q in ℕ_{0}, and any t in ]0, ∞[, we have that

(i) ψ_{q}^{c}(t) > 0.

(ii) (−1)^{q}ψ_{q}^{c}(−t) > 0.

(iii) |ψ_{q}^{c}(−t)| ≤ \frac{ψ_{0}^{c}(−t)}{ψ_{0}^{c}(t)}ψ_{q}^{c}(t).

Proof. (i) This follows from Lemma 7.3(i), but for completeness we include a different proof, which will also be needed in the proof of (ii) and (iii). For each q in ℕ_{0}, we put

ρ_{q}^{c}(x) = c^{− \frac{q}{2}} P_{q}^{c}(x), (x ∈ ℝ).

Then by Proposition 6.1, (ρ_{q}^{c})_{q∈ℕ_{0}} is an orthonormal basis for L_{2}([a, b], μ_{c}). Let A be the (bounded) operator for multiplication by x in L_{2}([a, b], μ_{c}). Then by (6.23) and (6.24), the matrix M(A) of A w.r.t. (ρ_{q}^{c})_{q∈ℕ_{0}}, is given by

M(A) = (\begin{matrix}c & \sqrt{c} & & & & 0 \\ \sqrt{c} & c + 1 & \sqrt{c} \\ & \sqrt{c} & c + 1 & \sqrt{c} \\ & & ... & ... & ... \\ 0 \end{matrix})  (7.22)

From this, it follows, that for any p in ℕ,

M(A^{p})_{jk} > 0, when |j − k| ≤ p,
M(A^{p})_{jk} = 0, when |j − k| > p.

Hence, for any t in [0, ∞[,

M(exp(tA))_{jk} = δ_{j,k} + ∑_{p=1}^{∞} \frac{t^{p}}{p!} M(A^{p})_{jk} > 0, (j, k ∈ ℕ_{0}).

Since exp(tA) is the operator for mutiplication by exp(tx) in L_{2}([a, b], μ_{c}), and since P_{0}^{c}(x) ≡ 1, we get that

ψ_{q}^{c}(t) = c^{−q} ∫_{a}^{b} exp(tx)P_{q}^{c}(x)P_{0}^{c}(x) dμ_{c}(x) = c^{− \frac{q}{2}} ⟨exp(tA)ρ_{q}^{c}, ρ_{0}^{c}⟩
= c^{− \frac{q}{2}} M(exp(tA))_{0,q} > 0,

(7.23)

and this proves (i).

(ii) To prove (ii), we consider the operator

B = A + 2P_{0},

where P_{0} is the projection onto ℂρ_{0}^{c} in B(L_{2}([a, b], μ_{c})). Then

M(B) = (\begin{matrix}c + 2 & \sqrt{c} & & & & 0 \\ \sqrt{c} & c + 1 & \sqrt{c} \\ & \sqrt{c} & c + 1 & \sqrt{c} \\ & & ... & ... & ... \\ 0 \end{matrix}) ,  (7.24)

so as above, we get that

M(exp(tB))_{jk} > 0, for all j, k in ℕ_{0}.  (7.25)

Let U be the unitary operator on L_{2}([a, b], μ_{c}), defined by the equation:

Uρ_{q}^{c} = (−1)^{q}ρ_{q}^{c}, (q ∈ ℕ_{0}).

Then

M(UBU^{∗}) = (\begin{matrix}c + 2 & −\sqrt{c} & & & & 0 \\ −\sqrt{c} & c + 1 & −\sqrt{c} \\ & −\sqrt{c} & c + 1 & −\sqrt{c} \\ & & ... & ... & ... \\ 0 \end{matrix}) = M(2(c + 1)111 − A).

Hence A = 2(c + 1)111 − UBU^{∗}, and for t in [0, ∞[, we have thus that

exp(−tA) = exp(−2(c + 1)t) exp(tUBU^{∗}) = exp(−2(c + 1)t)U exp(tB)U^{∗}.

Therefore,

M(exp(−tA))_{jk} = (−1)^{j+k} exp(−2(c + 1)t)M(exp(tB))_{jk},  (j, k ∈ ℕ_{0}),  (7.26)

so in particular, by (7.25),

(−1)^{j+k}M(exp(−tA))_{jk} > 0, (j, k ∈ ℕ_{0}).

For t in [0, ∞[, we note here that

ψ_{q}^{c}(−t) = c^{−q} ∫_{a}^{b} exp(−tx)P_{q}^{c}(x)P_{0}^{c}(x) dμ_{c}(x) = c^{− \frac{q}{2}} M(exp(−tA))_{q0},  (7.27)

and hence it follows that (−1)^{q}ψ_{q}(−t) > 0, which proves (ii).

To prove (iii), we need the following technical lemma:

7.6 Lemma. Let C and D be bounded positive selfadjoint operators on ℓ_{2}(ℕ_{0}), and assume that the corresponding matrices (c_{jk})_{j,k∈ℕ_{0}} and (d_{jk})_{j,k∈ℕ_{0}} satisfy the following conditions:

(a) c_{jk} ≥ 0 for all j, k in ℕ_{0}.
(b) c_{jk} = 0 when |j − k| ≥ 2.
(c) d_{jk} = c_{jk}, when (j, k) ≠ (0, 0).
(d) d_{00} ≥ c_{00}.

For φ, ψ in ℓ_{2}(ℕ_{0}), we define

[φ, ψ]_{j,k} = φ(j)ψ(k) − φ(k)ψ(j), (j, k ∈ ℕ_{0}).

Consider then furthermore f, g from ℓ_{2}(ℕ_{0}), satisfying that

(e) f(k) ≥ 0 and g(k) ≥ 0 for all k in ℕ_{0}.
(f) [f, g]_{j,k} ≥ 0, for all k, j in ℕ_{0} such that k > j.

Then for all j, k in ℕ_{0}, such that k > j, we have that

(i) [Cf, Cg]_{j,k} ≥ 0.
(ii) [Df, Cg]_{j,k} ≥ 0.
(iii) [D^{n}f, C^{n}g]_{j,k} ≥ 0, for all n in ℕ.
(iv) [exp(tD)f, exp(tC)g]_{j,k} ≥ 0, for all t in [0, ∞[.

7.7 Remark. If φ, ψ are strictly positive functions in ℓ_{2}(ℕ_{0}), then the statement

[φ, ψ]_{j,k} ≥ 0, for all j, k in ℕ_{0}, such that k > j,

is equivalent to the condition that

\frac{φ(0)}{ψ(0)} ≥ \frac{φ(1)}{ψ(1)} ≥ \frac{φ(2)}{ψ(2)} ≥ · · · . □

Proof of Lemma 7.6. Note first that for any φ, ψ in ℓ_{2}(ℕ_{0}) and j, k in ℕ_{0}, we have that [φ, ψ]_{j,k} = −[φ, ψ]_{k,j}. In particular,

[φ, ψ]_{j,j} = 0,  (φ, ψ ∈ ℓ_{2}(ℕ_{0}), j ∈ ℕ_{0}).  (7.28)

Note also that the positivity of C implies that

det (\begin{matrix}c_{jj} & c_{jk} \\ c_{kj} & c_{kk} \end{matrix}) ≥ 0,  for all j, k in ℕ_{0}, such that j ≠ k.  (7.29)

To prove (i), consider k, j in ℕ_{0}, such that k > j ≥ 0. We then have

(Cf)(j) = \begin{cases}c_{j,j−1}f(j − 1) + c_{j,j}f(j) + c_{j,j+1}f(j + 1), & if j ≥ 1, \\ c_{0,0}f(0) + c_{0,1}f(1), & if j = 0, \end{cases}

and since k ≠ 0,

(Cg)(k) = c_{k,k−1}g(k − 1) + c_{k,k}g(k) + c_{k,k+1}g(k + 1).

Thus,

[Cf, Cg]_{j,k} = \begin{cases}∑_{l=j−1}^{j+1} ∑_{m=k−1}^{k+1} c_{jl}c_{km}[f, g]_{l,m}, & if j ≥ 1, \\ ∑_{l=0}^{1} ∑_{m=k−1}^{k+1} c_{0l}c_{km}[f, g]_{l,m}, & if j = 0. \end{cases}

Assume first that k ≥ j + 2. In this case, l ≤ j + 1 ≤ k − 1 ≤ m, for all terms in the above sums, and thus, by (f) and (7.28), [f, g]_{l,m} ≥ 0. Since c_{lm} ≥ 0 for all l, m in ℕ_{0} (by (a)), it follows thus that [Cf, Cg]_{j,k} ≥ 0.

Assume next that k = j + 1, and consider first the case j ≥ 1. Then

[Cf, Cg]_{j,k} = ∑_{l=j−1}^{j+1} ∑_{m=j}^{j+2} c_{jl}c_{j+1,m}[f, g]_{l,m}.  (7.30)

In 8 of the 9 terms in the sum above, l ≤ m, and hence [f, g]_{l,m} ≥ 0. Only in the case (l, m) = (j + 1, j), do we have l > m. However, the sum of the two terms corresponding to (l, m) = (j, j + 1) and (l, m) = (j + 1, j) is non-negative, since

c_{jj}c_{j+1,j+1}[f, g]_{j,j+1} + c_{j,j+1}c_{j+1,j}[f, g]_{j+1,j} = (c_{jj}c_{j+1,j+1} − c_{j,j+1}c_{j+1,j})[f, g]_{j,j+1},

which is non-negative by (7.29). Since the remaining 7 terms in the sum on the right hand side of (7.30) are also non-negative, it follows that [Cf, Cg]_{j,k} ≥ 0. If j = 0, and k = j + 1 = 1, the same argument can be used to show that

[Cf, Cg]_{0,1} = ∑_{l=0}^{1} ∑_{m=0}^{2} c_{0l}c_{1m}[f, g]_{l,m} ≥ 0.

This proves (i).

To prove (ii), note first that by (a) and (c), we have

(Df)(j) = (Cf)(j), if j ≥ 1,

and

(Df)(0) = (Cf)(0) + (d_{00} − c_{00})f(0).

Hence, if k > j ≥ 1, we get from (i), that

[Df, Cg]_{j,k} = [Cf, Cg]_{j,k} ≥ 0.

If k > j = 0, then

[Df, Cg]_{0,k} = (Df)(0)(Cg)(k) − (Df)(k)(Cg)(0) = [Cf, Cg]_{0,k} + (d_{00} − c_{00})f(0)(Cg)(k).

But (d_{00} − c_{00})f(0) ≥ 0 by (d) and (e), and since also (Cg)(k) = ∑_{l=0}^{∞} c_{kl}g(l) ≥ 0, by (a) and (e), it follows by (i), that also [Df, Cg]_{0,k} ≥ 0. This proves (ii).

Next, (iii) follows from (ii) and induction on n, and from noting (by induction), that (D^{n}f)(j), (C^{n}g)(j) ≥ 0 for all n in ℕ and j in ℕ_{0}.

To prove (iv), we let t be a fixed number in [0, ∞[, and put

C_{n} = 111 + \frac{t}{n}C, and D_{n} = 111 + \frac{t}{n}D, (n ∈ ℕ_{0}).

Then, for all n, C_{n} and D_{n} are positive selfadjoint operators on ℓ_{2}(ℕ_{0}), which also satisfy the requirements (a)-(d). Hence, if f, g ∈ ℓ_{2}(ℕ_{0}) which satisfy (e) and (f), we conclude from (iii), that

[(111 + \frac{t}{n}D)^{n}f, (111 + \frac{t}{n}C)^{n}g]_{j,k} ≥ 0, when j > k,

and hence, letting n → ∞, we get that

[ exp(tD)f, exp(tC)g]_{j,k} ≥ 0, when j > k,

as desired. ■

End of Proof of Proposition 7.5. Only (iii) in Proposition 7.5 remains to be proved. Let A, B from B(L_{2}([a, b], μ_{c})) be as in the first part of the proof of Proposition 7.5. Since A is the multiplication operator associated to a positive function on [a, b], and since B ≥ A, both A and B are positive selfadjoint operators on L_{2}([a, b], μ_{c}). Let C and D be the operators in B(ℓ_{2}(ℕ_{0})) corresponding to A and B respectively, via the natural Hilbert space isomorphism between L_{2}([a, b], μ_{c}) and ℓ_{2}(ℕ_{0}), given by the orthonormal basis (ρ_{q}^{c})_{q∈ℕ_{0}} for L_{2}([a, b], μ_{c}). Then C and D are positive selfadjoint operators and by (7.22) and (7.24), they satisfy the conditions (a)-(d) of Lemma 7.6. Now, let both f and g be the first basis vector in the natural basis for ℓ_{2}(ℕ_{0}) (i.e., f(k) = g(k) = δ_{k,0} for all k in ℕ_{0}). Then (e),(f) of Lemma 7.6 are also satisfied, and hence we obtain from (iv) of that lemma, that for all j, k in ℕ_{0} such that k > j,

( exp(tD)f)(j)( exp(tC)f)(k) − ( exp(tD)f)(k)( exp(tC)f)(j) ≥ 0,

i.e.,

⟨ exp(tB)ρ_{0}^{c}, ρ_{j}^{c}⟩ · ⟨ exp(tA)ρ_{0}^{c}, ρ_{k}^{c}⟩ ≥ ⟨ exp(tB)ρ_{0}^{c}, ρ_{k}^{c}⟩ · ⟨ exp(tA)ρ_{0}^{c}, ρ_{j}^{c}⟩.

For j = 0, we get in particular,

\frac{M(exp(tB))_{k,0}}{M(exp(tA))_{k,0}} ≤ \frac{M(exp(tB))_{0,0}}{M(exp(tA))_{0,0}} ,  (k ∈ ℕ_{0}).  (7.31)

Note here, that by (7.26),

(−1)^{k}M(exp(−tA))_{k,0} = exp(−2(c + 1)t)M(exp(tB))_{k,0} > 0, (k ∈ ℕ_{0}).

Inserting this in (7.31), it follows that

\frac{(−1)^{k}M(exp(−tA))_{k,0}}{M(exp(tA))_{k,0}} ≤ \frac{M(exp(−tA))_{0,0}}{M(exp(tA))_{0,0}} ,  (k ∈ ℕ_{0}).  (7.32)

By (7.23) and (7.27),

M(exp(±tA))_{k,0} = c^{− \frac{k}{2}} ∫_{a}^{b} exp(±tx)P_{k}^{c}(x) dμ_{c}(x) = c^{\frac{k}{2}} ψ_{k}^{c}(±t), (k ∈ ℕ_{0}).

Hence, (iii) in Proposition 7.5 follows from (7.32). ■

7.8 Theorem. Let H and K be Hilbert spaces, and let a_{1}, …, a_{r} be elements of B(H, K) such that ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)}, and ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)}, for some constant c in [1, ∞[. Consider furthermore independent elements Y_{1}, …, Y_{r} of GRM(n, n, \frac{1}{n}), and put S = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}. Then for any t in [0, \frac{n}{2c}],

𝔼[ exp(−tS^{∗}S)] ≤ exp ( − (\sqrt{c} − 1)^{2}t + (c + 1)^{2} · \frac{t^{2}}{n} ) · 111_{B(H^{n})}.  (7.33)

Proof. Consider a fixed t in [0, \frac{n}{2c}]. By Proposition 7.4 and Proposition 7.5 we then have

\begin{aligned}∥𝔼[ exp(−tS^{∗}S)]∥ & ≤ ∑_{q=0}^{∞} |ψ_{q}^{c}(−t)| · ∥𝔼[P_{q}^{c}(S^{∗}S)]∥ \\ & ≤ \frac{ψ_{0}^{c}(−t)}{ψ_{0}^{c}(t)} ∑_{q=0}^{∞} ψ_{q}^{c}(t)∥𝔼[P_{q}^{c}(S^{∗}S)]∥. \end{aligned}  (7.34)

From (7.16) in the proof of Proposition 7.4, we have here that

\begin{aligned}∑_{q=0}^{∞} ψ_{q}^{c}(t) · ∥𝔼[P_{q}^{c}(S^{∗}S)]∥ & ≤ ∑_{p=0}^{∞} \frac{1}{p!}(\frac{ct}{n})^{p} ∑_{π∈S_{p}}n^{k(π̂)}(\frac{n}{c})^{l(π̂)−1} \\ & ≤ ∑_{p=0}^{∞} \frac{1}{p!}(\frac{ct}{n})^{p} ∑_{\begin{matrix}k,l∈ℕ \\ k+l≤p+1 \end{matrix}}δ(p, k, l)n^{k}(\frac{n}{c})^{l−1}, \end{aligned}

where δ(p, k, l) was introduced in (3.6). Applying now Lemma 3.6, we get for t in [0, \frac{n}{2c}], that

∑_{q=0}^{∞} ψ_{q}^{c}(t) · ∥𝔼[P_{q}^{c}(S^{∗}S)]∥ ≤ exp ((n + \frac{n}{c})(\frac{ct}{n})^{2})∫_{a}^{b} exp (\frac{n}{c}(\frac{ct}{n}x)) dμ_{c}(x)

≤ exp ((c + 1)^{2} · \frac{t^{2}}{n} )∫_{a}^{b} exp(tx) dμ_{c}(x).

Note here, that ψ_{0}^{c}(t) = ∫_{a}^{b} exp(tx) dμ_{c}(x), and hence we get by (7.34), that

∥𝔼[ exp(−tS^{∗}S)]∥ ≤ exp ((c + 1)^{2} · \frac{t^{2}}{n} )ψ_{0}^{c}(−t)
= exp ((c + 1)^{2} · \frac{t^{2}}{n} )∫_{a}^{b} exp(−tx) dμ_{c}(x).

But exp(−tx) ≤ exp(−ta) = exp(−t(\sqrt{c} + 1)^{2}) for all x in [a, b], and hence it follows that

∥𝔼[ exp(−tS^{∗}S)]∥ ≤ exp ((c + 1)^{2} · \frac{t^{2}}{n} ) exp(−(\sqrt{c} − 1)^{2}t), (t ∈ [0, \frac{n}{2c}]).

This proves (7.33). ■

7.9 Remark. By application of the method of Remark 3.7, it is easy to extend Theorem 7.8, to the case where

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)},  and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} = d111_{B(K)},

for constants c, d such that c ≥ d > 0. In this case, one obtains that for t in [0, \frac{n}{2c}],

𝔼[ exp(−tS^{∗}S)] ≤ exp ( − (\sqrt{c} − \sqrt{d})^{2}t + (c + d)^{2} · \frac{t^{2}}{n} ) · 111_{B(H^{n})}. □

8 Asymptotic Lower Bound on the Spectrum of S_{n}^{∗}S_{n} in the Exact Case

Let H and K be Hilbert spaces, and consider elements a_{1}, …, a_{r} of B(H, K). Let A denote the C^{∗}-subalgebra of B(H), generated by the family {a_{i}^{∗}a_{j} | i, j ∈ {1, 2, …, r}}. Consider furthermore, for each n in ℕ, independent elements Y_{1}^{(n)}, …, Y_{r}^{(n)} of GRM(n, n, \frac{1}{n}), and define

S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)},  (n ∈ ℕ).  (8.1)

In this section, we shall determine (almost surely), the asymptotic behaviour of the smallest element of the spectrum of S_{n}^{∗}S_{n}, under the assumptions that A is an exact C^{∗}-algebra and that a_{1}, …, a_{r} satisfy the condition

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ 111_{B(K)},  (8.2)

for some constant c in [1, ∞[. We start, however, by considering the simpler case, where, instead of (8.2), a_{1}, …, a_{r}, satisfy the stronger condition

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(K)},  (8.3)

for some constant c in [1, ∞[. Once this simpler case has been handled, we obtain the more general case by virtue of a dilation result.

As in Section 4, we determine first the asymptotic behaviour of the smallest eigenvalue of V_{n}, where

V_{n} = (Φ ⊗ id_{n})(S_{n}^{∗}S_{n}),  (n ∈ ℕ),  (8.4)

and Φ: A → M_{d}(ℂ) is a completely positive mapping, for some d in ℕ.

8.1 Lemma. Let S_{n}, n ∈ ℕ, and V_{n}, n ∈ ℕ, be as in (8.1) and (8.4), and assume that a_{1}, …, a_{r} satisfy the condition (8.3). Let λ_{min}(V_{n}) denote the smallest eigenvalue of V_{n} (considered as an element of M_{dn}(ℂ)). Then for any ϵ in ]0, ∞[, we have that

∑_{n=1}^{∞} P (λ_{min}(V_{n}) ≤ (\sqrt{c} − 1)^{2} − ϵ) < ∞.

Proof. The proof is basically the same as the proof of Lemma 4.2; the main difference being that in this proof we apply Theorem 7.8 instead of Theorem 3.3. Consequently, we shall not repeat all details in this proof.

For fixed n in ℕ, and arbitrary t in ]0, ∞[, we find that

P (λ_{min}(V_{n}) ≤ (\sqrt{c} − 1)^{2} − ϵ) = P ( exp ( − tλ_{min}(V_{n}) + t(\sqrt{c} − 1)^{2} − tϵ) ≥ 1)
≤ exp(t(\sqrt{c} − 1)^{2} − tϵ) · 𝔼[ exp ( − tλ_{min}(V_{n}))]
≤ exp(t(\sqrt{c} − 1)^{2} − tϵ) · 𝔼 ∘ Tr_{dn}[ exp(−tV_{n})].

(8.5)

By application of Lemma 4.1(ii), we have here, that

tr_{dn}[ exp(−tV_{n})] = tr_{dn}[ exp ( − t(Φ ⊗ id_{n})(S_{n}^{∗}S_{n}))] ≤ tr_{dn}[(Φ ⊗ id_{n})(exp(−tS_{n}^{∗}S_{n}))]
= tr_{d} ⊗ tr_{n}[(Φ ⊗ id_{n})(exp(−tS_{n}^{∗}S_{n}))] = ϕ ⊗ tr_{n}[ exp(−tS_{n}^{∗}S_{n})],

(8.6)

where ϕ is the state tr_{d} ∘ Φ on A. It follows here from Definition 3.1 and Theorem 7.8, that

𝔼[ϕ ⊗ tr_{n}( exp(−tS_{n}^{∗}S_{n}))] = ϕ ⊗ tr_{n}(𝔼[ exp(−tS_{n}^{∗}S_{n})])

≤ exp ( − t(\sqrt{c} − 1)^{2} + \frac{t^{2}}{n} (c + 1)^{2}), (8.7)

for all t in ]0, \frac{n}{2c}]. Combining now (8.5)-(8.7), it follows that for all t in ]0, \frac{n}{2c}],

P (λ_{min}(V_{n}) ≤(\sqrt{c} − 1)^{2} − ϵ)
≤ dn · exp(t(\sqrt{c} − 1)^{2} − tϵ) · exp ( − t(\sqrt{c} − 1)^{2} + \frac{t^{2}}{n} (c + 1)^{2})
= dn · exp (t(\frac{t}{n}(c + 1)^{2} − ϵ)).

From here, the proof is concluded exactly as the proof of Theorem 4.2. ■

8.2 Proposition. Let S_{n}, n ∈ ℕ, and V_{n}, n ∈ ℕ, be as in (8.1) and (8.4), and assume that a_{1}, …, a_{r} satisfy the condition (8.3). We then have

lim inf_{n→∞} λ_{min}(V_{n}) ≥ (\sqrt{c} − 1)^{2}, almost surely.

Proof. By Lemma 4.2 and the Borel-Cantelli Lemma (cf. [Bre, Lemma 3.14]), we have for any ϵ from ]0, ∞[, that

P (λ_{min}(V_{n}) ≥ (\sqrt{c} − 1)^{2} − ϵ, for all but finitely many n) = 1,

and from this the proposition follows readily. ■

The next two lemmas enable us to pass from the situation considered in Proposition 8.2 to the more general situation, where it is only assumed that a_{1}, …, a_{r} satisfy (8.2).

8.3 Lemma. Let c be a number in [1, ∞[, and put q = 2 + [c], where [c] denotes the integer part of c. Then there exist elements x_{1}, …, x_{q} in the Cuntz algebra O_{2}, such that

∑_{i=1}^{q} x_{i}^{∗}x_{i} = c111_{O_{2}},  and  ∑_{i=1}^{q} x_{i}x_{i}^{∗} = 111_{O_{2}}.

Proof. Recall that O_{2} is the unital C^{∗}-algebra C^{∗}(s_{1}, s_{2}) generated by two operators s_{1}, s_{2} satisfying that s_{i}^{∗}s_{j} = δ_{i,j}111_{O_{2}}, i, j ∈ {1, 2}, and that s_{1}s_{1}^{∗} + s_{2}s_{2}^{∗} = 111_{O_{2}}. Define then t_{1}, …, t_{q−1} in O_{2}, by the expression

t_{j} = \begin{cases}s_{2}^{j−1}s_{1}, & if j ∈ \{1, 2, …, q − 2\}, \\ s_{2}^{q−2}, & if j = q − 1. \end{cases}

Then t_{i}^{∗}t_{j} = δ_{i,j}111_{O_{2}}, for all i, j in {1, 2, …, q − 1}, and

∑_{j=1}^{q−1} t_{j}t_{j}^{∗} = ∑_{i=0}^{q−3} s_{2}^{i}(111_{O_{2}} − s_{2}s_{2}^{∗})(s_{2}^{i})^{∗} + s_{2}^{q−2}(s_{2}^{q−2})^{∗} = 111_{O_{2}},  (8.8)

(i.e., t_{1}, …, t_{q−1} generates a copy of O_{q−1} inside O_{2}). Define now x_{1}, …, x_{q} in O_{2}, by

x_{i} = \begin{cases}(\frac{c − 1}{q − 2})^{\frac{1}{2}} t_{i}, & if i ∈ \{1, 2, …, q − 1\} \\ (\frac{q − 1 − c}{q − 2})^{\frac{1}{2}}111_{O_{2}}, & if i = q. \end{cases}

Then

∑_{i=1}^{q} x_{i}^{∗}x_{i} = (q − 1) · \frac{c − 1}{q − 2} · 111_{O_{2}} + \frac{q − 1 − c}{q − 2} · 111_{O_{2}} = c111_{O_{2}},

and by (8.8),

∑_{i=1}^{q} x_{i}x_{i}^{∗} = \frac{c − 1}{q − 2} · 111_{O_{2}} + \frac{q − 1 − c}{q − 2} · 111_{O_{2}} = 111_{O_{2}}.

Thus, x_{1}, …, x_{q} have the desired properties. ■

8.4 Lemma. Let H and K be Hilbert spaces, and let a_{1}, …, a_{r} be elements of B(H, K), such that ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)}, and ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ 111_{B(K)}.

Then there exist Hilbert spaces H̃, K̃, s in {r, r + 1, r + 2, … } and elements ã_{1}, …, ã_{s} of B(H̃, K̃), such that the following conditions hold:

(i) H̃ ⊇ H and K̃ ⊇ K.

(ii) ã_{i}_{|H} = \begin{cases}a_{i}, & if 1 ≤ i ≤ r, \\ 0, & if r + 1 ≤ i ≤ s. \end{cases}

(iii) ∑_{i=1}^{s} ã_{i}^{∗}ã_{i} = c111_{B(H̃)} and ∑_{i=1}^{s} ã_{i}ã_{i}^{∗} = 111_{B(K̃)}.

Proof. By Lemma 8.3, we may choose finitely many elements x_{1}, …, x_{q} of the Cuntz algebra O_{2}, such that ∑_{i=1}^{q} x_{i}^{∗}x_{i} = c111_{O_{2}} and ∑_{i=1}^{q} x_{i}x_{i}^{∗} = 111_{O_{2}}. We assume that O_{2} is represented on some Hilbert space L, so that x_{1}, …, x_{r} ∈ B(L). Define then

H̃ = (H ⊗ L) ⊕ (K ⊗ L) and K̃ = (K ⊗ L) ⊕ (H ⊗ L).

For Hilbert spaces V, W, an element v of B(V, W), and an element y of B(L), we consider v ⊗ y as an element of B(V ⊗ L, W ⊗ L) in the natural manner. Moreover, given v_{11} in B(H ⊗ L, K ⊗ L), v_{12} in B(K ⊗ L), v_{21} in B(H ⊗ L) and v_{22} in B(K ⊗ L, H ⊗ L), we shall consider the matrix (v_{ij})_{1≤i,j≤1} as an element of B(H̃, K̃) in the usual way. With these conventions, consider now the following elements of B(H̃, K̃),

\begin{aligned}ã_{i} & = (\begin{matrix}a_{i} ⊗ 111_{B(L)} & 0 \\ 0 & 0 \end{matrix}) ,  (i ∈ \{1, 2, …, r\}), \\ b_{j} & = (\begin{matrix}0 & (111_{B(K)} − ∑_{i=1}^{r} a_{i}a_{i}^{∗})^{\frac{1}{2}} ⊗ x_{j} \\ 0 & 0 \end{matrix}) ,  (j ∈ \{1, 2, …, q\}), \\ c_{i,j,k} & = (\begin{matrix}0 & 0 \\ 0 & \frac{1}{\sqrt{c}} · a_{i}^{∗} ⊗ (x_{j}x_{k}) \end{matrix}) ,  (i ∈ \{1, 2, …, r\}, j, k ∈ \{1, 2, …, q\}). \end{aligned}

It follows then by direct calculation, that

\begin{matrix}∑_{i=1}^{r} & ã_{i}^{∗}ã_{i} + ∑_{j=1}^{q} b_{j}^{∗}b_{j} + ∑_{i=1}^{r} ∑_{j,k=1}^{q} c_{i,j,k}^{∗}c_{i,j,k} \\ = (\begin{matrix}[ ∑_{i=1}^{r} a_{i}^{∗}a_{i}] ⊗ 111_{B(L)} & 0 \\ 0 & [c(111_{B(K)} − ∑_{i=1}^{r} a_{i}a_{i}^{∗}) + c ∑_{i=1}^{r} a_{i}a_{i}^{∗}] ⊗ 111_{B(L)} \end{matrix}) = c111_{B(H̃)}, \end{matrix}

and that

\begin{matrix}∑_{i=1}^{r} & ã_{i}ã_{i}^{∗} + ∑_{j=1}^{q} b_{j}b_{j}^{∗} + ∑_{i=1}^{r} ∑_{j,k=1}^{q} c_{i,j,k}c_{i,j,k}^{∗} \\ = (\begin{matrix}[ ∑_{i=1}^{r} a_{i}a_{i}^{∗} + (111_{B(K)} − ∑_{i=1}^{r} a_{i}a_{i}^{∗})] ⊗ 111_{B(L)} & 0 \\ 0 & [\frac{1}{c} ∑_{i=1}^{r} a_{i}^{∗}a_{i}] ⊗ 111_{B(L)} \end{matrix}) = 111_{B(K̃)}. \end{matrix}

Thus, if we put s = r + q + rq^{2}, and let ã_{r+1}, ã_{r+2}, …, ã_{s}, be new names for the elements in the set {b_{j} | j ∈ {1, …, q}} ∪ {c_{i,j,k} | i ∈ {1, …, r}, j, k ∈ {1, …, q}}, then it follows that ã_{1}, ã_{2}, …, ã_{s} satisfy condition (iii).

Choosing a fixed unit vector ξ in L, we have natural embeddings ι_{H}: H → H̃ and ι_{K}: K → K̃ given by the equations

\begin{aligned}ι_{H}(h) & = (h ⊗ ξ) ⊕ 0,  (h ∈ H), \\ ι_{K}(k) & = (k ⊗ ξ) ⊕ 0,  (k ∈ K). \end{aligned}

This justifies (i), and moreover, it is straightforward to check, that under the identifications of H with ι_{H}(H) and K with ι_{K}(K), condition (ii) is satisfied. This concludes the proof. ■

8.5 Proposition. Let S_{n}, n ∈ ℕ, and V_{n}, n ∈ ℕ, be as in (8.1) and (8.4), and assume now that a_{1}, …, a_{r} satisfy the condition (8.2). Then

lim inf_{n→∞} λ_{min}(V_{n}) ≥ (\sqrt{c} − 1)^{2}, almost surely.

Proof. By Lemma 8.4, we may choose Hilbert spaces H̃, K̃, s in {r, r + 1, …, } and elements ã_{1}, ã_{2}, …, ã_{s} of B(H, K), such that conditions (i)-(iii) of Lemma 8.4 are satisfied. If r < s, then for each n in ℕ we choose additional elements Y_{r+1}^{(n)}, …, Y_{s}^{(n)} of GRM(n, n, \frac{1}{n}), such that Y_{1}^{(n)}, Y_{2}^{(n)}, …, Y_{s}^{(n)} are independent. We then define

S̃_{n} = ∑_{i=1}^{s} ã_{i} ⊗ Y_{i}^{(n)}, (n ∈ ℕ).

Recall from (8.4), that

V_{n} = (Φ ⊗ id_{n})(S_{n}^{∗}S_{n}), (n ∈ ℕ),

where Φ: A → M_{d}(ℂ) is a completely positive mapping from the C^{∗}-subalgebra A of B(H) generated by {a_{i}^{∗}a_{j} | i, j ∈ {1, 2, …, r}}, into the matrix algebra M_{d}(ℂ). By [Pa, Theorem 5.2], there exists a completely positive mapping Φ_{1} : B(H) → M_{d}(ℂ) extending Φ. Note that since Φ is unital, so is Φ_{1}.

Consider next the orthogonal projection P_{H} of H̃ onto H. Then the mapping

C_{P_{H}}: b ↦ P_{H}bP_{H}: B(H̃) → P_{H}B(H̃)P_{H} ≃ B(H),

is unital completely positive. Hence, so is the mapping Φ_{2} : B(H̃) → M_{d}(ℂ), given by

Φ_{2}(b) = Φ_{1}(P_{H}bP_{H}) = Φ_{1} ∘ C_{P_{H}}(b), (b ∈ B(H̃)).

Thus, if we define

Ṽ_{n} = (Φ_{2} ∘ id_{n})(S̃_{n}^{∗}S̃_{n}), (n ∈ ℕ),

then it follows from Lemma 8.4(iii) and Proposition 8.2, that

lim inf_{n→∞} λ_{min}(Ṽ_{n}) ≥ (\sqrt{c} − 1)^{2},  almost surely.  (8.9)

However, by Lemma 8.4(ii), we have here that

Ṽ_{n} = (Φ_{2} ⊗ id_{n})[ ∑_{i,j=1}^{s} ã_{i}^{∗}ã_{j} ⊗ (Y_{i}^{(n)})^{∗}Y_{j}^{(n)}] = ∑_{i,j=1}^{s} Φ_{2}(ã_{i}^{∗}ã_{j}) ⊗ (Y_{i}^{(n)})^{∗}Y_{j}^{(n)}

= ∑_{i,j=1}^{s} Φ_{1}(P_{H}ã_{i}^{∗}ã_{j}P_{H}) ⊗ (Y_{i}^{(n)})^{∗}Y_{j}^{(n)} = ∑_{i,j=1}^{r} Φ_{1}(a_{i}^{∗}a_{j}) ⊗ (Y_{i}^{(n)})^{∗}Y_{j}^{(n)}

= ∑_{i,j=1}^{r} Φ(a_{i}^{∗}a_{j}) ⊗ (Y_{i}^{(n)})^{∗}Y_{j}^{(n)} = V_{n}.

Therefore (8.9) yields the desired conclusion. ■

It remains now to show that we can replace V_{n} in Proposition 8.5 by S_{n}^{∗}S_{n} itself. Before proceeding with this task, we draw attention to the following simple observation:

8.6 Lemma. For each n in ℕ, let B_{n} be a unital C^{∗}-algebra, and let b_{n} be an element of B_{n}. Then for any R in [0, ∞[, the following two conditions are equivalent:

(i) lim sup_{n→∞}∥b_{n}∥ ≤ R.

(ii) lim sup_{n→∞} max(sp(b_{n})) ≤ R, and lim inf_{n→∞} min(sp(b_{n})) ≥ −R.

Proof. This is clear, since, for each n, ∥b_{n}∥ is the largest of the two numbers max(sp(b_{n})) and − min(sp(b_{n})). ■

8.7 Theorem. Let a_{1}, …, a_{r} be elements of B(H, K), such that ∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ 111_{B(K)}, for some constant c in [1, ∞[. Assume, in addition, that the unital C^{∗}- subalgebra A of B(H), generated by the set {a_{i}^{∗}a_{j} | i, j, ∈ {1, 2, …, r}}, is exact. Consider furthermore, for each n in ℕ, independent elements Y_{1}^{(n)}, …, Y_{r}^{(n)} of GRM(n, n, \frac{1}{n}), and put S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}, n ∈ ℕ. We then have

lim inf_{n→∞} min [sp(S_{n}^{∗}S_{n})] ≥ (\sqrt{c} − 1)^{2},  almost surely.  (8.10)

Proof. Put E = span{a_{i}^{∗}a_{j} | i, j ∈ {1, 2, …, r}}, and note that x^{∗} ∈ E for all x in E, and that 111_{A} = c^{−1} ∑_{i=1}^{r} a_{i}^{∗}a_{i} ∈ E. Thus, E is a finite dimensional operator system, and since A is exact, it follows thus from Proposition 4.4, that for any ϵ from ]0, ∞[, there exist d in ℕ and a unital completely positive mapping Φ: A → M_{d}(ℂ), such that

∥(Φ ⊗ id_{n})(x)∥ ≥ (1 − ϵ)∥x∥,  (n ∈ ℕ, x ∈ M_{n}(E)).  (8.11)

Consider now a fixed ϵ from ]0, ∞[, let d, Φ be as described above, and define

V_{n} = (Φ ⊗ id_{n})(S_{n}^{∗}S_{n}), (n ∈ ℕ).

Recall then from Proposition 4.3 and Proposition 8.5, that

lim sup_{n→∞} max [sp(V_{n})] ≤ c + 1 + 2\sqrt{c},  almost surely,
lim inf_{n→∞} min [sp(V_{n})] ≥ c + 1 − 2\sqrt{c},  almost surely,

and hence that

lim sup_{n→∞} max [sp(V_{n} − (c + 1)111_{dn})] ≤ 2\sqrt{c},  almost surely,
lim inf_{n→∞} min [sp(V_{n} − (c + 1)111_{dn})] ≥ −2\sqrt{c},  almost surely.

By Lemma 8.6, this means that

lim sup_{n→∞} ∥V_{n} − (c + 1)111_{dn}∥ ≤ 2\sqrt{c},  almost surely.  (8.12)

Note here, that since S_{n}^{∗}S_{n} − (c + 1)111_{A⊗M_{n}(ℂ)} ∈ M_{n}(E), for all n, it follows from (8.11), that

\begin{aligned}∥S_{n}^{∗}S_{n} − (c + 1)111_{A⊗M_{n}(ℂ)}∥ & ≤ (1 − ϵ)^{−1}∥(Φ ⊗ id_{n})[S_{n}^{∗}S_{n} − (c + 1)111_{A⊗M_{n}(ℂ)}]∥ \\ & = (1 − ϵ)^{−1}∥V_{n} − (c + 1)111_{dn}∥, \end{aligned}

for all n in ℕ. Hence (8.12) implies that

lim sup_{n→∞} ∥S_{n}^{∗}S_{n} − (c + 1)111_{A⊗M_{n}(ℂ)}∥ ≤ (1 − ϵ)^{−1} · 2\sqrt{c}, almost surely.

Since this holds for arbitrary ϵ from ]0, ∞[, it follows that actually

lim sup_{n→∞} ∥S_{n}^{∗}S_{n} − (c + 1)111_{A⊗M_{n}(ℂ)}∥ ≤ 2\sqrt{c}, almost surely.

By Lemma 8.6, this implies, in particular, that

lim inf_{n→∞} min [sp(S_{n}^{∗}S_{n}) − (c + 1)] ≥ −2\sqrt{c}, almost surely,

and this proves (8.10). ■

8.8 Remark. As for the upper bound (cf. Section 4), Theorem 8.7 does not, in general, hold without the condition, that the C^{∗}-algebra generated by {a_{i}^{∗}a_{j} | 1 ≤ i, j ≤ r} be exact. In fact, for any c in ]1, ∞[, it is possible to choose a finite set of elements a_{1}, …, a_{r} of B(H), for an infinite dimensional Hilbert space H, such that

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{B(H)} and ∑_{i=1}^{r} a_{i}a_{i}^{∗} = 111_{B(H)},

but at the same time

P ( 0 ∈ sp(S_{n}^{∗}S_{n}), for all but finitely many n) = 1,

where S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}, as in (8.1). The proof of this is, however, much more complicated than the corresponding proof of the possible violation for the upper bound (cf. Proposition 4.9(ii)), and it will be presented elsewhere. □

9 Comparison of Projections in Exact C^{∗}-algebras and states on the K_{0}-group

In [Haa], the first named author proved that quasitraces on exact, unital C^{∗}-algebras are traces. This result implies the following two theorems

9.1 Theorem. (cf. [Han], [Haa]) If A is an exact, unital, stably finite C^{∗}-algebra, then A has a tracial state.

9.2 Theorem. (cf. [BR, Corollary 3.4]) If A is an exact, unital C^{∗}-algebra, then every state on K_{0}(A) comes from a tracial state on A.

The proof given in [Haa] of the fact that quasitraces in exact unital C^{∗}-algebras are traces, is based on an ultra-product argument, involving ultra products of finite AW^{∗}-algebras.

The aim of this section is to show that Theorem 9.1 and Theorem 9.2 can be obtained from the random matrix results of the previuos sections, without appealing to results on quasitraces and AW^{∗}-algebras.

We start by recapturing some of the standard notions and notation in connection with comparison theory for projections in C^{∗}-algebras. For a C^{∗}-algebra A, we put

M_{∞}(A) = ⋃_{n∈ℕ}M_{n}(A),

where elements are indentified via the (non-unital) embeddings M_{n}(A) ↪ M_{n+1}(A), given by addition of a row and a column of zeroes. Given two projections p, q in M_{∞}(A), we say, as usual, that p and q are (Murray-von Neumann) equivalent, and write p ∼ q, if there exists a u in M_{∞}(A), such that u^{∗}u = p and uu^{∗} = q. We let V (A) denote the set of equivalence classes ⟨p⟩ of projections p in M_{∞}(A), w.r.t. Murray-von Neumann equivalence, and we equip V (A) with an order structure and an addition, as follows: For projections p, q in M_{∞}(A), we write ⟨q⟩ ≤ ⟨p⟩ if q ≺ p, i.e., if q is equivalent to a subprojection of p. Moreover, we define ⟨p⟩+⟨q⟩ to be ⟨p′ +q′⟩, where p′, q′ are projections in M_{∞}(A), satisfying that p′ ∼ p, q′ ∼ q and p′ ⊥ q′. Finally, for k in ℕ, we let k⟨p⟩ denote the equivalence class ⟨p⟩ + ⋯ + ⟨p⟩ (k summands).

Recall that K_{0}(A) is the additive group obtained from the semi group V (A), via the Grothendieck construction (cf. [Bl1]), and that K_{0}(A)_{+} denotes the range of V (A) under the natural map

ρ: V (A) → K_{0}(A).

In particular, we have that K_{0}(A) = K_{0}(A)_{+} − K_{0}(A)_{+}.

For a projection p in M_{∞}(A), we put

[p] = ρ(⟨p⟩).

Note then, that for projections p, q in M_{∞}(A), [p] = [q] if and only if there exists a projection r in M_{∞}(A), such that ⟨p⟩ + ⟨r⟩ = ⟨q⟩ + ⟨r⟩.

9.3 Lemma. Let A be a C^{∗}-algebra, and let p, q be projections in A. Then with I(p) the ideal in A generated by p, the following three conditions are equivalent:

(i) ⟨q⟩ ≤ k⟨p⟩, for some k in ℕ.

(ii) q ∈ I(p).

(iii) q ∈ \overline{I(p)}.

Proof. (i) ⇒ (ii) : Assume that (i) holds, i.e., that there exists k in ℕ and u in M_{k}(A), such that

u^{∗}u = (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) and uu^{∗} ≤ (\begin{matrix}p & & & & 0 \\ & . & . & . \\ 0 & & & & p \end{matrix}) .

This implies that u is of the form

u = (\begin{matrix}u_{11} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ u_{k1} & 0 & ⋯ & 0 \end{matrix}) ,

where u_{11}, u_{21}, …, u_{k1} ∈ pAq. It follows thus, that

q = ∑_{j=1}^{k} u_{j1}^{∗}u_{j1} = ∑_{j=1}^{k} u_{j1}^{∗}pu_{j1} ∈ I(p),

as desired.

(ii) ⇒ (iii) : This is trivial.

(iii) ⇒ (i) : Assume that (iii) holds. Then there exist k in ℕ and a_{1}, …, a_{k}, b_{1}, …, b_{k} in A, such that

∥ ∑_{j=1}^{k} a_{j}pb_{j} − q∥ ≤ \frac{1}{2}.  (9.1)

Furthermore, we may assume that pb_{j}q = b_{j} for all j (otherwise, substitute b_{j} by pb_{j}q), and that A is a C^{∗}-subalgebra of B(H) for some Hilbert space H. From (9.1), it follows then that

∥( ∑_{j=1}^{k}(a_{j}pb_{j})ξ) − ξ∥ ≤ \frac{1}{2}∥ξ∥, for all ξ in q(H).

Hence, with t = max {∥a_{j}∥ | j ∈ {1, 2, …, k}}, we have that

t ∑_{j=1}^{k} ∥(pb_{j})ξ∥ ≥ ∥ ∑_{j=1}^{k}(a_{j}pb_{j})ξ∥ ≥ \frac{1}{2}∥ξ∥, (ξ ∈ q(H)),

which by the Cauchy-Schwarz inequality implies that

t\sqrt{k}( ∑_{j=1}^{k} ∥pb_{j}ξ∥^{2})^{\frac{1}{2}} ≥ \frac{1}{2}∥ξ∥,  (ξ ∈ q(H)).  (9.2)

Since b_{j}q = b_{j} for all j, we have that ∑_{j=1}^{k} b_{j}^{∗}pb_{j} ∈ qAq, and hence (9.2) implies that

t^{2}k ∑_{j=1}^{k} b_{j}^{∗}pb_{j} ≥ \frac{1}{4}q.

Thus, ∑_{j=1}^{k} b_{j}^{∗}pb_{j} is invertible in the unital C^{∗}-algebra qAq. Let h ∈ qAq be the inverse of ∑_{j=1}^{k} b_{j}^{∗}pb_{j} in qAq, and put c_{j} = b_{j}h^{\frac{1}{2}} , for all j. Then since pb_{j}q = b_{j} for all j, pc_{j}q = c_{j} for all j, and hence

∑_{j=1}^{k} c_{j}^{∗}c_{j} = ∑_{j=1}^{k} c_{j}^{∗}pc_{j} = h^{\frac{1}{2}} ( ∑_{j=1}^{k} b_{j}^{∗}pb_{j})h^{\frac{1}{2}} = q.

Define now

u = (\begin{matrix}c_{1} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ c_{k} & 0 & ⋯ & 0 \end{matrix}) ∈ M_{k}(A).

Then

u^{∗}u = (\begin{matrix}∑_{j=1}^{k} c_{j}^{∗}c_{j} & 0 \\ 0 & 0 \end{matrix}) = (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) ,

and

uu^{∗} = (c_{i}c_{j}^{∗})_{1≤i,j≤k} ∈ M_{k}(pAp).

Since u^{∗}u is a projection, so is uu^{∗}. Thus uu^{∗} is a projection in the C^{∗}-algebra M_{k}(pAp), and hence uu^{∗} ≤ p ⊗ 111_{k}. Taken together, we have verified that

(\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) = u^{∗}u ∼ uu^{∗} ≤ p ⊗ 111_{k},

which shows that (i) holds. ■

9.4 Lemma. Let M be a von Neumann algebra, and let p be a projection in M. Then any σ-weakly lower semi-continuous trace

τ: (pMp)_{+} → [0, ∞],

has an extension to a σ-weakly lower semi-continuous trace τ̃ on M_{+}.

Proof. We can assume that p ≠ 0. Choose then a maximal family (p_{i})_{i∈I} of pairwise orthogonal projections in M, such that p_{i} ≺ p for all i in I. Then, by standard comparison theory, it follows that

∑_{i∈I} p_{i} = c(p),

where c(p) denotes the central support of p in M. Choose next, for each i in I, a partial isometry v_{i} in M, such that

v_{i}^{∗}v_{i} = p_{i} and v_{i}v_{i}^{∗} ≤ p, (i ∈ I).

Define then τ̃: M_{+} → [0, ∞], by the equation

τ̃(a) = ∑_{i∈I} τ(v_{i}av_{i}^{∗}), (a ∈ M_{+}).

Clearly τ̃ is additive, homogenous and σ-weakly lower semi-continuous. To show that τ̃ has the trace property, note first that since pv_{i} = v_{i} for all i, we have also that c(p)v_{i} = v_{i} for all i. Since c(p) is in the center of M, it follows thus, that for any x in M,

\begin{aligned}τ̃(xx^{∗}) & = ∑_{i∈I} τ(v_{i}xx^{∗}v_{i}^{∗}) = ∑_{i∈I} τ(c(p)v_{i}xx^{∗}v_{i}^{∗}) \\ & = ∑_{i∈I} τ(v_{i}xc(p)x^{∗}v_{i}^{∗}) = ∑_{i∈I} ∑_{j∈I} τ((v_{i}xv_{j}^{∗})(v_{j}x^{∗}v_{i}^{∗})), \end{aligned}

and similarly

τ̃(x^{∗}x) = ∑_{j∈I} ∑_{i∈I} τ((v_{j}x^{∗}v_{i}^{∗})(v_{i}xv_{j}^{∗})).

But by the trace property of τ on pMp, we have that

τ((v_{i}xv_{j}^{∗})(v_{j}x^{∗}v_{i}^{∗})) = τ((v_{j}x^{∗}v_{i}^{∗})(v_{i}xv_{j}^{∗})),

for all i, j, and since all the terms in the above sums are positive, we can permute their order without changing the sums, and thus obtain

τ̃(xx^{∗}) = τ̃(x^{∗}x).

Taken together, we have verified that τ̃ is a σ-weakly lower semi-continuous trace on M_{+}, and it remains thus to show that τ̃ coincides with τ on (pMp)_{+}. Given a from (pMp)_{+}, we have that v_{i}a^{\frac{1}{2}} ∈ pMp, for all i, and therefore

τ̃(a) = ∑_{i∈I} τ((v_{i}a^{\frac{1}{2}} )(a^{\frac{1}{2}} v_{i}^{∗})) = ∑_{i∈I} τ(a^{\frac{1}{2}} v_{i}^{∗}v_{i}a^{\frac{1}{2}} ) = τ(a^{\frac{1}{2}} c(p)a^{\frac{1}{2}} ) = τ(a),

as desired. ■

9.5 Lemma. Let M be a von Neumann algebra, and let 111 denote the unit of M. Let furthermore p, q be projections in M, that satisfy the following two conditions:

(i) 111 ∈ I(p).

(ii) τ(q) ≤ τ(p), for any normal, tracial state τ on M.

Then q ≺ p.

Proof. Let M = eM ⊕ (111 − e)M, be the decomposition of M into a finite part eM and a properly infinite part (111 − e)M, by a central projection e. Since any normal, tracial state on M must vanish on (111 − e)M, condition (ii) is equivalent to the condition

τ(eq) ≤ τ(ep), for any normal tracial state τ on eM.

By comparison theory for finite von Neumann algebras (cf. e.g. [KR, Theorem 8.4.3(vii)]), this condition implies that

eq ≺ ep in eM,  (9.3)

By Lemma 9.3, condition (i) implies that there exists a k in ℕ, such that

111 ⊗ e_{11} ≺ p ⊗ 111_{k} in M_{k}(M),

where (e_{ij})_{1≤i,j≤k} are the usual matrix units in M_{k}(ℂ). Therefore, we have also that

(111 − e) ⊗ e_{11} ≺ (111 − e)p ⊗ 111_{k} in M_{k}((111 − e)M).

At the same time, since 111 − e is a properly infinite projection in M, we have that

(111 − e) ⊗ e_{11} ∼ (111 − e) ⊗ 111_{k} in M_{k}((111 − e)M).

It follows thus, that

(111 − e)q ⊗ 111_{k} ≤ (111 − e) ⊗ 111_{k} ∼ (111 − e) ⊗ e_{11} ≺ (111 − e)p ⊗ 111_{k} in M_{k}((111 − e)M),

and by [KR, Exercise 6.9.14], this implies that

(111 − e)q ≺ (111 − e)p in (111 − e)M.  (9.4)

Combining (9.3) and (9.4), it follows that q ≺ p, as desired. ■

9.6 Lemma. Let M be a von Neumann algebra, and let p, q be projections in M. Then the following two conditions are equivalent

(i) q ≺ p.

(ii) q ∈ I(p), and τ(q) ≤ τ(p) for every σ-weakly lower semi-continuous trace τ on M_{+}.

Proof. Clearly (i) implies (ii). To show that (ii) implies (i), assume that (ii) holds. By Lemma 9.3 there exists then a k in ℕ, such that ⟨q⟩ ≤ k⟨p⟩, i.e., such that

q ⊗ e_{11} ∼ q′ ≤ p ⊗ 111_{k},

for some projection q′ in M_{k}(M). Consider now the von Neumann algebra

N = M_{k}(pMp),

with unit 111_{N} = p ⊗ 111_{k}. Set p′ = p ⊗ e_{11}. Then p′, q′ are both projections in N, and

111_{N} ∈ I_{N} (p′),  (9.5)

where I_{N} (p′) is the ideal in N generated by p′.

We show next, that

τ(q′) ≤ τ(p′), for any normal, tracial state τ on N .  (9.6)

Indeed, if τ is a normal, tracial state on N, then by Lemma 9.4, the restriction τ_{|N_{+}} of τ to N_{+} can be extended to a σ-weakly lower semi-continuous trace τ̃ on M_{k}(M)_{+}. Then the mapping

a ↦ τ̃(a ⊗ e_{11}), (a ∈ M_{+}),

is a σ-weakly lower semi-continuous trace on M_{+}, and hence the assumption (ii) yields that

τ̃(q ⊗ e_{11}) ≤ τ̃(p ⊗ e_{11}).

Since q′ ∼ q ⊗ e_{11}, p′ = p ⊗ e_{11} and p′, q′ ∈ N, it follows thus that

τ(q′) = τ̃(q′) = τ̃(q ⊗ e_{11}) ≤ τ̃(p ⊗ e_{11}) = τ̃(p′) = τ(p′),

which proves (9.6).

Applying now Lemma 9.5, it follows from (9.5) and (9.6), that q′ ≺ p′ in N, and hence that

q ⊗ e_{11} ∼ q′ ≺ p′ = p ⊗ e_{11} in M_{k}(M),

which implies that q ≺ p in M. ■

9.7 Proposition. Let A be a C^{∗}-algebra, and let p, q be projections in A. Then the following two conditions are equivalent:

(i) q ≺ p in A^{∗∗}.

(ii) τ(q) ≤ τ(p), for every (norm) lower semi-continuous trace τ on A_{+}.

Proof. (i) ⇒ (ii) : Assume that q ≺ p in A^{∗∗}, and choose u in A^{∗∗}, such that u^{∗}u = q and uu^{∗} ≤ p. Then ∥u∥ ≤ 1, and hence by the Kaplansky Density Theorem, we may choose a net (u_{β})_{β∈B} from A, such that

∥u_{β}∥ ≤ 1, for all β in B, and u_{β} → u in the strong (operator) topology.

Define now: v_{β} = pu_{β}q, (β ∈ B), and note that v_{β} → puq = u in the strong (operator) topology, so that v_{β}^{∗}v_{β} → u^{∗}u = q in the weak (operator) topology. Since ∥v_{β}∥ ≤ 1 for all β, this implies that actually

v_{β}^{∗}v_{β} → q in the σ-weak topology.

Note also, that since ∥u_{β}∥ ≤ 1 for all β,

v_{β}v_{β}^{∗} ≤ p,  (β ∈ B).  (9.7)

Recall now that the σ-weak topologi on A^{∗∗} is the weak* topology i.e., the σ(A^{∗∗}, A^{∗})- topology, and hence its restriction to A is the weak topology, i.e., the σ(A, A^{∗})-topology. Since v_{β} ∈ A for all β, we have thus, that

v_{β}^{∗}v_{β} → q in the σ(A, A^{∗})-topology.

Consider then the convex hull K of {v_{β}^{∗}v_{β} | β ∈ B}. Then q ∈ K^{−σ(A,A^{∗})}, but since convex sets in a Banach space have the same closure in weak and norm topology (cf. [KR, Theorem 1.3.4]), it follows that actually q ∈ K^{−norm}. Hence, we may choose a sequence (w_{n})_{n∈ℕ} from K, which converges to q in norm. Then, for any (norm) lower semi-continuous trace τ: A_{+} → [0, ∞],

τ(q) ≤ lim inf_{n→∞} τ(w_{n}) ≤ sup_{β∈B} τ(v_{β}^{∗}v_{β}) = sup_{β∈B} τ(v_{β}v_{β}^{∗}) ≤ τ(p),  (9.8)

and this proves (i).

(ii) ⇒ (i) : Assume (ii) holds. We set out to show that condition (ii) in Lemma 9.6 is satisfied, in the case M = A^{∗∗}. Consider first the function τ_{0} : A_{+} → [0, ∞], defined by

τ_{0}(a) = \begin{cases}0, & if a ∈ \overline{I_{A}(p)_{+}}, \\ ∞, & if a ∈ A_{+} \ \overline{I_{A}(p)_{+}}. \end{cases}

Then τ_{0} is a (norm) lower semi-continuous trace on A_{+}, and hence the assumption (ii) yields that τ_{0}(q) ≤ τ_{0}(p) = 0, which means that q ∈ \overline{I_{A}(p)_{+}}. According to Lemma 9.3, this implies that actually q ∈ I_{A}(p) ⊆ I_{A^{∗∗}}(p).

Note next, that for any σ-weakly lower semi-continuous trace τ on (A^{∗∗})_{+}, the restriction τ_{|A_{+}} is a (norm) lower semi-continuous trace on A, and hence, by the assumption (ii), τ(q) ≤ τ(p).

Taken together, we have verified that the projections p, q satisfy the condition (ii) in Lemma 9.6, in the case M = A^{∗∗}, and hence this lemma yields that q ≺ p in A^{∗∗}, as desired. ■

9.8 Corollary. Let A be a C^{∗}-algebra, and let p, q be projections in A. Then the following two conditions are equivalent:

(i) ∃k ∈ ℕ: k⟨q⟩ ≤ (k − 1)⟨p⟩ in V (A^{∗∗}).

(ii) ∃ϵ > 0: τ(q) ≤ (1 − ϵ)τ(p), for any (norm) lower semi-continuous trace τ on A_{+}.

Proof. (i) ⇒ (ii) : Assume that (i) holds, and define, for the existing k, q′ = q ⊗ 111_{k} and p′ = p ⊗ ( ∑_{i=1}^{k−1} e_{ii}). Then q′, p′ are projections in M_{k}(A), and the assumption (i) implies that

q′ ≺ p′ in M_{k}(A^{∗∗}).  (9.9)

Given now any (norm) lower semi-continuous trace τ on A_{+}, note that the expression

τ_{k}(a) = ∑_{i=1}^{k} τ(a_{ii}), (a = (a_{ij}) ∈ M_{k}(A)_{+}),

defines a (norm) lower semi-continuous trace τ on M_{k}(A)_{+}. Thus, by Proposition 9.7, (9.9) implies that τ_{k}(q′) ≤ τ_{k}(p′), i.e., that kτ(q) ≤ (k − 1)τ(p). This shows that (ii) holds for any ϵ in ]0, \frac{1}{k}].

(ii) ⇒ (i) : Assume that (ii) holds, and choose, for the existing ϵ, a k in ℕ such that \frac{1}{k} ≤ ϵ. Define then, for this k, q′ and p′ as above.

Now, for any (norm) lower semi-continuous trace τ on M_{k}(A)_{+}, the mapping

a ↦ τ(a ⊗ e_{11}), (a ∈ A_{+}),

is a (norm) lower semi-continuous trace on A_{+}, and thus the assumption (ii) yields that

τ(q ⊗ e_{11}) ≤ (1 − ϵ)τ(p ⊗ e_{11}) ≤ \frac{k−1}{k} · τ(p ⊗ e_{11}),

and hence that

τ(q′) = k · τ(q ⊗ e_{11}) ≤ (k − 1) · τ(p ⊗ e_{11}) = τ(p′).

According to Proposition 9.7, this means that q′ ≺ p′ in M_{k}(A^{∗∗})(= M_{k}(A)^{∗∗}), which shows that (i) holds. ■

9.9 Lemma. Let A be a C^{∗}-algebra, and let p, q be projections in A. Then the following two conditions are equivalent:

(i) There exists an ϵ in ]0, ∞[, such that

τ(q) ≤ (1 − ϵ)τ(p), for any (norm) lower semi-continuous trace τ on A_{+}.

(ii) There exist ϵ in ]0, ∞[, r in ℕ and a_{1}, …, a_{r} in A, such that

∑_{i=1}^{r} a_{i}^{∗}a_{i} = q,  and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ (1 − ϵ)p.

Proof. The proof follows the ideas of the first section of [Haa].

Note first that (ii) clearly implies (i). To show the converse implication, assume that (i) holds. Then, by Corollary 9.8, there exists a k in ℕ, such that

q ⊗ 111_{k} ≺ p ⊗ (∑_{i=1}^{k−1}e_{ii}) in M_{k}(A^{∗∗}),

i.e., such that

u^{∗}u = q ⊗ 111_{k},  and uu^{∗} ≤ p ⊗ (∑_{i=1}^{k−1}e_{ii}),  (9.10)

for some u = (u_{ij})_{1≤i,j≤k} in M_{k}(A^{∗∗}). For this u, we have then that

∑_{j=1}^{k} ∑_{i=1}^{k} u_{ij}^{∗}u_{ij} = ∑_{j=1}^{k}(u^{∗}u)_{jj} = kq,

and that

∑_{i=1}^{k} ∑_{j=1}^{k} u_{ij}u_{ij}^{∗} = ∑_{i=1}^{k}(uu^{∗})_{ii} ≤ (k − 1)p.

Thus, if b_{1}, …, b_{k^{2}} ∈ A^{∗∗} denote the elements \frac{1}{\sqrt{k}}u_{ij}, i, j ∈ \{1, 2, …, k\}, listed in any fixed order, then we have that

∑_{i=1}^{k^{2}} b_{i}^{∗}b_{i} = q,  and  ∑_{i=1}^{k^{2}} b_{i}b_{i}^{∗} ≤ \frac{k−1}{k}p.

Note also, that (9.10) implies that b_{i} ∈ pA^{∗∗}q for all i. Consider then the subset K of A ⊕ A, defined by

K = {(∑_{i=1}^{r} c_{i}^{∗}c_{i}, g + ∑_{i=1}^{r} c_{i}c_{i}^{∗}) | r ∈ ℕ, c_{1}, …, c_{r} ∈ pAq, g ∈ (pAp)_{+}}.

Then K is clearly closed under addition and multiplication by a non-negative scalar, and thus K is a convex cone in A ⊕ A.

Recall next, that the σ-strong^{∗} topology on a von Neumann algebra M, is generated by the semi-norms

x ↦ φ(x^{∗}x + xx^{∗})^{\frac{1}{2}} , (φ ∈ (M_{∗})_{+}).

Since the σ-strong^{∗} continuous functionals on M are also σ-weakly continuous (i.e., belong to M_{∗}; cf. [Ta, Lemma II.2.4]), any convex set in M has the same closure in σ-strong^{∗} and σ-weak topology. In particular it follows that

pAq is σ-strong^{∗} dense in pA^{∗∗}q, and (pAp)_{+} is σ-strong^{∗} dense in (pA^{∗∗}p)_{+}.

Thus, we may choose a net (c_{1}^{α}, …, c_{k^{2}}^{α}, g^{α})_{α∈A} in [ ⊕_{j=1}^{k^{2}} pAq] ⊕ (pAp), such that

• c_{i}^{α} → b_{i}, in the σ-strong^{∗} topology, for all i in {1, 2, …, k^{2}},
• g^{α} ≥ 0, for all α,
• g^{α} → \frac{k−1}{k}p − ∑_{i=1}^{k^{2}} b_{i}b_{i}^{∗}, in the σ-strong^{∗} topology.

It follows then that

lim_{α} ( ∑_{i=1}^{k^{2}}(c_{i}^{α})^{∗}c_{i}^{α}) = q, σ-weakly,

and that

lim_{α} (g^{α} + ∑_{i=1}^{k^{2}} c_{i}^{α}(c_{i}^{α})^{∗}) = \frac{k−1}{k}p, σ-weakly.

But since the σ-weak topology on A^{∗∗} is just the weak^{∗}-topology (i.e., the σ(A^{∗∗}, A^{∗})- topology), its restriction to A is the weak topology (i.e., the σ(A, A^{∗})-topology) on A. It follows thus that

(q, \frac{k−1}{k}p) ∈ K^{−σ(A⊕A,A^{∗}⊕A^{∗})}.

But convex sets in a Banach space have the same closure in weak and norm topology (cf. [KR, Theorem 1.3.4]), so it follows that in fact

(q, \frac{k−1}{k}p) ∈ K^{−norm}.  (9.11)

Since (1 − δ)^{−1}(\frac{k−1}{k} + δ) → \frac{k−1}{k} < 1, as δ → 0, we may choose δ, ϵ in ]0, 1[, such that

(1 − δ)^{−1}(\frac{k−1}{k} + δ) = 1 − ϵ.

By (9.11), there exist then r in ℕ, c_{1}, …, c_{r} in pAq and g in (pAp)_{+}, such that

∥q − ∑_{i=1}^{r}c_{i}^{∗}c_{i}∥ < δ and ∥\frac{k−1}{k}p − g − ( ∑_{i=1}^{r} c_{i}c_{i}^{∗})∥ < δ.  (9.12)

The first inequality in (9.12) implies that ∑_{i=1}^{r} c_{i}^{∗}c_{i} is invertible in the C^{∗}-algebra qAq. Let h ∈ (qAq)_{+} denote the inverse of ∑_{i=1}^{r} c_{i}^{∗}c_{i} in qAq. Since

(1 − δ)q ≤ ∑_{i=1}^{r} c_{i}^{∗}c_{i} ≤ (1 + δ)q,

it follows then that

(1 + δ)^{−1}q ≤ h ≤ (1 − δ)^{−1}q.  (9.13)

Define now: a_{i} = c_{i}h^{\frac{1}{2}} , i ∈ \{1, 2, …, r\}. Then ∑_{i=1}^{r} a_{i}^{∗}a_{i} = q, and moreover, by (9.13) and the second inequality in (9.12),

\begin{aligned}∑_{i=1}^{r} a_{i}a_{i}^{∗} & = ∑_{i=1}^{r} c_{i}hc_{i}^{∗} ≤ (1 − δ)^{−1} ∑_{i=1}^{r} c_{i}c_{i}^{∗} ≤ (1 − δ)^{−1}(g + ∑_{i=1}^{r} c_{i}c_{i}^{∗}) \\ & ≤ (1 − δ)^{−1}(\frac{k−1}{k} + δ)p = (1 − ϵ)p. \end{aligned}

Thus, it follows that (ii) holds. ■

9.10 Theorem. Let A be an exact C^{∗}-algebra, and let p, q be projections in A. Assume that there exists ϵ in ]0, ∞[, such that

τ(q) ≤ (1 − ϵ)τ(p), for any (norm) lower semi-continuous trace τ on A_{+}.

Then there exists n in ℕ, such that

q ⊗ 111_{n} ≺ p ⊗ 111_{n} in M_{n}(A).

Proof. By Lemma 9.9, we get (after multiplying the a_{i}’s from Lemma 9.9(ii) by (1−ϵ)^{− \frac{1}{2}} ), that there exist c in ]1, ∞[, r in ℕ and a_{1}, …, a_{r} in A, such that

∑_{i=1}^{r} a_{i}^{∗}a_{i} = cq,  and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ p.  (9.14)

We may assume that A is a C^{∗}-subalgebra of B(H) for some Hilbert space H. Then (9.14) implies that we may consider a_{1}, …, a_{r} as elements of B(q(H), p(H)), and that

∑_{i=1}^{r} a_{i}^{∗}a_{i} = c111_{q(H)},  and  ∑_{i=1}^{r} a_{i}a_{i}^{∗} ≤ 111_{p(H)}.

Moreover, the set {a_{i}^{∗}a_{j} | i, j ∈ {1, 2, …, r}} is contained in the exact, unital C^{∗}- algebra qAq. Choosing now, for each n in ℕ, independent elements Y_{1}^{(n)}, …, Y_{r}^{(n)} of GRM(n, n, \frac{1}{n}), it follows from Theorem 8.7, that with

S_{n} = ∑_{i=1}^{r} a_{i} ⊗ Y_{i}^{(n)}, (n ∈ ℕ),

we have that

lim inf_{n→∞} [ min \{sp(S_{n}(ω)^{∗}S_{n}(ω))\}] ≥ (\sqrt{c} − 1)^{2}, for almost all ω in Ω.

In particular, there exsits one(!) ω in Ω, and an n in ℕ, such that S_{n}(ω)^{∗}S_{n}(ω) is invertible in the C^{∗}-algebra M_{n}(qAq). For this pair (ω, n), we define

u = S_{n}(ω)[S_{n}(ω)^{∗}S_{n}(ω)]^{− \frac{1}{2}} ,

where the inverse is formed w.r.t. M_{n}(qAq). Then u ∈ M_{n}(pAq), and

u^{∗}u = 111_{q(H)} ⊗ 111_{n} = q ⊗ 111_{n}.  (9.15)

Moreover, uu^{∗} ∈ M_{n}(B(p(H))), and since u^{∗}u is a projection in M_{n}(B(q(H))), uu^{∗} is a projection in M_{n}(B(p(H))), so that

uu^{∗} ≤ 111_{p(H)} ⊗ 111_{n} = p ⊗ 111_{n}.  (9.16)

Combining (9.15) and (9.16), we obtain the desired conclusion. ■

9.11 Corollary. If A is an exact, unital and simple C^{∗}-algebra, and p, q are projections in A, such that p ≠ 0 and τ(q) < τ(p) for all tracial states τ on A, then for some n in ℕ

q ⊗ 111_{n} ≺ p ⊗ 111_{n} in M_{n}(A).  (9.17)

Proof. By simplicity of A, τ(p) > 0 for all tracial states τ on A, and hence by weak* compactness of the set of tracial states on A, there exists ϵ in ]0, ∞[, such that

τ(q) ≤ (1 − ϵ)τ(p),

for all tracial states τ on A. By the assumptions on A, A is algebraicly simple. Hence, every non-zero trace τ: A_{+} → [0, ∞] is either equal to +∞ on all of A_{+}\{0}, or proportional to a tracial state. Hence we can apply Theorem 9.10. ■

9.12 Remark. In the “inequality” (9.17) in Corollary 9.11, the tensoring with 111_{n} can in general not be avoided. This follows from Villadsen’s result in [Vi] that there exist nuclear (and hence exact) unital simple C^{∗}-algebras with perforation. Recall that a unital C^{∗}-algebra A has perforation, if there exists x in K_{0}(A), such that x ∉ K_{0}(A)_{+}, but nx ∈ K_{0}(A)_{+} \ {0}, for some n in ℕ. To see how Villadsen’s result implies, that we cannot, in general, avoid tensoring with 111_{n} in (9.17), let A be a unital exact simple C^{∗}- algebra, and assume that x ∈ K_{0}(A), such that x ∉ K_{0}(A)_{+} and nx ∈ K_{0}(A)_{+} \ {0}, for some positive integer n. Write then x in the form x = [p] − [q], where p, q are projections in M_{k}(A) for some k in ℕ. By the assumption that nx ∈ K_{0}(A)_{+} \{0}, and the simplicity of A, it is not hard to deduce that

(τ ⊗ tr_{k})(p) > (τ ⊗ tr_{k})(q),

for all tracial states τ on A, and hence τ̃(p) > τ̃(q) for all tracial states τ̃ on M_{k}(A). However, since x ∉ K_{0}(A)_{+}, q cannot be equivalent to a subprojection of p. □

9.13 Theorem. Let A be a unital, exact C^{∗}-algebra. Then the following two conditions are equivalent:

(i) A has no tracial states.

(ii) For some n in ℕ there exist projections p, q in M_{n}(A), such that

p ⊥ q and p ∼ q ∼ 111_{A} ⊗ 111_{n}.

Proof. Clearly, (ii) implies (i). To show the converse implication, assume that (i) holds, and consider then the two projections p′, q′ in M_{2}(A) given by

p′ = (\begin{matrix}111_{A} & 0 \\ 0 & 0 \end{matrix}) , and q′ = (\begin{matrix}111_{A} & 0 \\ 0 & 111_{A} \end{matrix}) .

Since A has no tracial states, A^{∗∗} has no normal tracial states, and hence A^{∗∗} is a properly infinite von Neumann algebra. Therefore,

⟨111_{A}⟩ = 4⟨111_{A}⟩ in V (A^{∗∗}),

which implies that

⟨p′⟩ = 2⟨q′⟩ in V (M_{2}(A^{∗∗})).

Hence by Corollary 9.8 and Theorem 9.10, there exists an n in ℕ, such that

q′ ⊗ 111_{n} ≺ p′ ⊗ 111_{n} in M_{2n}(A).

Here, p′ ⊗ 111_{n} ∼ (\begin{matrix}111_{A} ⊗ 111_{n} & 0 \\ 0 & 0 \end{matrix}), and thus there exists u in M_{2n}(A), such that

u^{∗}u = (\begin{matrix}111_{A} ⊗ 111_{n} & 0 \\ 0 & 111_{A} ⊗ 111_{n} \end{matrix}) ,  and uu^{∗} ≤ (\begin{matrix}111_{A} ⊗ 111_{n} & 0 \\ 0 & 0 \end{matrix}) .  (9.18)

The inequality in (9.18) implies that u has the form

u = (\begin{matrix}u_{11} & u_{12} \\ 0 & 0 \end{matrix}) ,

for suitable u_{11}, u_{12} from M_{n}(A). The equality in (9.18) yields then subsequently that

u_{11}^{∗}u_{11} = u_{12}^{∗}u_{12} = 111_{A} ⊗ 111_{n}, and u_{11}^{∗}u_{12} = 0.

Defining now

p = u_{11}u_{11}^{∗} and q = u_{12}u_{12}^{∗},

it follows that p, q are orthogonal projections in M_{n}(A), such that p ∼ q ∼ 111_{A} ⊗ 111_{n}. This shows that (ii) holds. ■

In particular, Theorem 9.15 implies the validity of Theorem 9.1:

9.14 Corollary. If A is an exact, unital, stably finite C^{∗}-algebra, then A has a tracial state.

Proof. This is an obvious consequence of Theorem 9.13. ■

Consider next an arbitrary unital C^{∗}-algebra A. A function φ: V (A) → ℝ, is said to be a state on V (A), if it satisfies the following three conditions:

• φ(x) ≥ 0, for all x in V (A).
• φ(x + y) = φ(x) + φ(y), for all x, y in V (A).
• φ(⟨111_{A}⟩) = 1.

We denote by S(V (A)), the set of states on V (A). It is elementary to check that S(V (A)) is a convex set, and that S(V (A)) is compact in “the topology of pointwise convergence”.

9.15 Lemma. Let A be a unital, exact C^{∗}-algebra, and let p, q be projections in A, such that

τ(q) ≤ τ(p), for any tracial state τ on A.  (9.19)

Then for any k in ℕ, there exists n in ℕ, such that

nk⟨q⟩ ≤ nk⟨p⟩ + n⟨111_{A}⟩.

Proof. Let k from ℕ be given, and consider then the projections p′, q′ in M_{k+1}(A) defined by:

p′ = p ⊗ (∑_{i=1}^{k} e_{ii}) + 111_{A} ⊗ e_{k+1,k+1}, and q′ = q ⊗ (∑_{i=1}^{k} e_{ii}).

Given now an arbitrary non-zero, bounded trace τ on M_{k+1}(A), note that the mapping

a ↦ τ(a ⊗ e_{11}), (a ∈ A),

is proportional to a tracial state on A. It follows thus from the assumption (9.19), that τ(q ⊗ e_{11}) ≤ τ(p ⊗ e_{11}), and hence

τ(q′) = k · τ(q ⊗ e_{11}) ≤ k · τ(p ⊗ e_{11}) = \frac{k}{k+1} · τ(p ⊗ 111_{k+1}) ≤ \frac{k}{k+1} · τ(p′).

Since 111_{A} ⊗ e_{11} ≺ p′, any unbounded (lower semi-continuous) trace τ on M_{k+1}(A) must take the value +∞ at p′, and hence we have also in this case, that

τ(q′) ≤ \frac{k}{k+1} · τ(p′).

Applying now Theorem 9.10, it follows that there exists an n in ℕ, such that n⟨q′⟩ ≤ n⟨p′⟩, and hence such that nk⟨q⟩ ≤ nk⟨p⟩ + n⟨111_{A}⟩, as desired. ■

Next, we need the following version of the Goodearl-Handelman theorem (see [Bl2, 3.4.7], [Go, 7.11] and [Rø, Lemma 2.9]).

9.16 Lemma. Let A be a unital C^{∗}-algebra, and consider a convex subset K of S(V (A)), which is closed in “the topology of pointwise convergence”. Assume furthermore that the following implication holds

∀x, y ∈ V (A): [∀φ ∈ K: φ(x) ≤ φ(y)] ⟹ [∀φ ∈ S(V (A)): φ(x) ≤ φ(y)]. (9.20)

Then K = S(V (A)).

Proof. The lemma is just a small variation of [Rø, Lemma 2.9]; the only difference being that in [Rø, Lemma 2.9], the condition (9.20) is replaced by the following

∀x, y ∈ V (A): [∀φ ∈ K: φ(x) < φ(y)] ⟹ [∀φ ∈ S(V (A)): φ(x) < φ(y)]. (9.21)

It suffices thus to show, that among the conditions (9.20) and (9.21) on K, (9.20) is the stronger.

So assume that (9.20) holds, and consider elements x, y of V (A), satisfying that

φ(x) < φ(y), for all φ in K.

Then since S(V (A)), and hence K, is compact in “the topology of weak convergence”, it follows that there exists a k in ℕ, such that

φ(x) + \frac{1}{k} ≤ φ(y), for all φ in K,

i.e., such that

φ(kx + ⟨111_{A}⟩) ≤ φ(ky), for all φ in K.

Applying then (9.20), it follows that for all φ in S(V (A)),

φ(x) = \frac{1}{k} · φ(kx) < \frac{1}{k} · φ(kx + ⟨111_{A}⟩) ≤ \frac{1}{k} · φ(ky) = φ(y),

which proves that (9.21) holds. ■

9.17 Theorem. Let A be a unital, exact C^{∗}-algebra. Then for any state φ on V (A), there exists a tracial state τ on A, such that

φ(⟨p⟩) = (τ ⊗ Tr_{m})(p), for all projections p in M_{m}(A), and m in ℕ.  (9.22)

Proof. Let K denote the subset of S(V (A)) consisting of those states on V (A), that are given by (9.22) for some tracial state τ on A. Then K is clearly a convex, compact subset of S(V (A)), and hence, by Lemma 9.16, it suffices to verify that K satisfies condition (9.20). So consider projections p, q in M_{∞}(A). We may assume that p, q ∈ M_{m}(A), for some sufficiently large positive integer m. Suppose then that

(τ ⊗ Tr_{m})(q) ≤ (τ ⊗ Tr_{m})(p), for all tracial states τ on A.

Since any tracial state on M_{m}(A) has the form \frac{1}{m} · τ ⊗ Tr_{m}, for some tracial state τ on A, it follows then from Lemma 9.15, that for any k in ℕ, there exists an n in ℕ, such that

nk⟨q⟩ ≤ nk⟨p⟩ + n⟨111_{A} ⊗ 111_{m}⟩.

Hence for any φ in S(V (A)), and any k in ℕ, we have that

φ(⟨q⟩) ≤ φ(⟨p⟩) + \frac{m}{k},

and this shows that K satisfies condition (9.20). ■

Recall that a state on K_{0}(A) is an additive map ψ: K_{0}(A) → ℝ, such that ψ(K_{0}(A)_{+}) ⊆ [0, ∞[ and ψ([111_{A}]) = 1. Note that the natural map ρ: V (A) → K_{0}(A), gives a one-to-one correspondance between states on V (A) and states on K_{0}(A). Hence, Theorem 9.17 gives a new proof, not relying on quasitraces, for the following

9.18 Corollary. Let A be a unital, exact C^{∗}-algebra. Then any state on K_{0}(A) comes from a tracial state on A.

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Department of Mathematics and Computer Science Odense University Campusvej 55, 5230 Odense M Denmark

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