Abstract: Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain ``discrete Fourier transform'' of a random variable. This provides a simple unified method to understand the known examples of cumulants, like free cumulants and various q-cumulants.
Abstract: We continue the investigation of noncommutative cumulants. In this paper various characterizations of noncommutative Gaussian random variables are proved.
Abstract: Fock space constructions give rise to natural exchangeable families and are thus well suited for cumulant calculations. In this paper we develop some general formulas and compute cumulants for generalized Toeplitz operators, notably for q-Fock spaces, previously considered by M. Anshelevich and A. Nica.
Abstract: A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.
Abstract: A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected set partitions respectively. The proof relies on Moebius inversion on the partition lattice.
Abstract: We use free probability techniques to compute borders of spectra of non hermitian operators in finite von Neumann algebras which arise as "free sums" of "simple" operators. To this end, the resolvent is analyzed with the aid of the Haagerup inequalidty. Concrete examples coming from reduced C*-algebras of free product groups and leading to systems of polynomial equations illustrate the approach.
Abstract: We use free probability techniques for computing spectra and Brown measures of some non hermitian operators in finite von Neumann algebras. Examples include where and are the generators of Zn and Z respectively, in the free product Zn*Z, or elliptic elements, of the form where and are free semi-circular elements of variance and .
Abstract: Let be the generators of the free group and denote by its left regular representation. A formula for norms of operators of the form where are complex matrices or operators is derived similar to the formula of Akemann and Ostrand for the scalar case. This formula can be used to compute norms of arbitrary finitely supported convolution operators on the free group. Essentially the same technique yields a formula for the norms of sums of matrix valued creation and annihilation operators on full Fock space. In a final section, some results about numerical evaluation of the formula are proved.
Abstract: Let G be a discrete group and denote by its left regular representation on . Denote further by the free group on n generators and its left regular representation. In this paper we show that a subset of G has the Leinert property if and only if for some real positive coefficients the identity holds. Using the same method we obtain some metric estimates about abstract unitaries satisfying the similar identity
Abstract: Let be the generators of the free group and denote by its left regular representation. An inequality for norms of operators of the form (for special ) similar to the formula of Akemann and Ostrand for the scalar case is obtained, however an example shows that strict inequality can occur. As a corollary we get the optimal constant in an inequality of Haagerup and Pisier.
Preface (102 Kb)
Chapter 1: M_n spaces (197 Kb)
Chapter 2: sums of unitaries and harmonic analysis on the free group (175 Kb)
Abstract The first part of the thesis investigates -spaces and their connections with operator spaces and Banach spaces. An -space is defined as a Banach space X together with a norm on the space of n times n matrices with entries in X satisfying some Ruan-type conditions. We develop a theory inspired by Blecher's and Paulsen's results on operator spaces including dual spaces and maximal and minimal operator space structures associated to an -space. Another section is dedicated to certain norms on tensor products of -spaces, which we call reasonable, where analogues of the Haagerup tensor product generalize the injective and projective tensor products of Banach spaces.
In the second part of the thesis we investigate a recent inequality for the norms of sums of certain unitaries. The combinatorial principle behind this inequality turns out to be the concept of nonnegative alternating mixed moments, which is studied in the first section. We then show that in the case of convolution operators on discrete groups equality takes place in this inequality if and only if the support of the operator in consideration satisfies the Leinert property, generalizing a result of Kesten. We give a counterexample to a conjectured generalization of this result about the structure of unitaries for which the above inequality turns into an equality, involving the representations of the free group constructed by Pytlik and Szwarc. However, we show some inequalities for subfamilies of families of unitaries whose sum attains minimal norm. Finally, an inequality for the norms of matrix-valued free operators is derived, which yields the optimal constant in an inequality by Haagerup and Pisier.