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Noncommutative Distributions by Sergio Albeverio download in pdf, ePub, iPad

Dyson equation that

Voiculescu and formulas for integration over unitary groups of B. We will also indicate some results on Brown measures of some sums and products of free random variables. The study of characters on infinite groups was initiated by Thoma in where he classified extreme characters on the group of finite permutations of the natural numbers. Popa, Mihai Queen's University Hardy classes on some non-commutative unit balls The talk will present some results, joint work with V. We will also discuss the immediate consequences of this work.

This is joint work with Serban Belinschi and Ion Nechita. We will then discuss a couple of constructions which can be used to produce new examples of such exchangeable brownian motions. Anderson provides in many situations and particularly in free probability theory a powerful tool to deal with polynomials in non-commutative variables. Such variables, like the quantum mechanical observables are operators and the distributions are expectations associated with them. The talk is on joint work with Uffe Haagerup.

More generally the free gamma distributions

In the talk I will report on a recent detailed study of the free Gamma distributions based on the method of Stieltjes inversion. This is joint work with G. An attempt will be made to describe our results and this hurdle as well as the reasons for our interest in Wielandt's theorem. Our methods combine the general theory of non-commutative functions with asymptotic freeness results of D. Absolute continuity, tail behavior and unimodality.

More generally the free gamma distributions are defined similarly in terms of the mentioned bijection. Dyson equation that a specific formal power series in non-commuting variables is algebraic. Whereas in the one-dimensional case it is abelian this is no longer true in dimensions bigger or equal to two. We will start with those origins, then report the current status of the study of those quantities, and finally conclude a few questions. This is joint work with Darren Creutz.

We propose a way to drop this delocalization assumption and we generalize this theorem to a quite general framework, including random projections whose corresponding vectors are localized, i. This has some important consequences, which we shall present, in particular also by examples from the analytic category. Further we show that these classes are closed under this type of amalgamated free product. The classification of these objects has recently been completed in joint work with Sven Raum, and I will report on that.

This is joint work with B. In this talk we will describe a similar rigidity property for characters on many irreducible lattices in higher rank semi-simple groups. The results are part of a larger work on non-commutative symmetric domains. The solution to this problem clarifies and improves the violation of additivity for minimum output entropy in quantum information theory. These are compact matrix quantum groups with a very intrinsic combinatorial structure, given by partitions.

The approach is further extended to the multiple interval case, by taking into account the multivalued nature of the inverse of the Cauchy transform. This is joint work with D. In the case of one variable the noncommutative distributions are expectations of spectral measures and are classical, i. We finally describe some applications related to the free Segal-Bargmann transform introduced by Biane.