Regularity in Free Probability and Free Stein Dimension

Ian Charlesworth (Cardiff University)

Thursday 30th March, 2023 16:00-17:00 Maths 311B


The study of regularity in free probability boils down to the question of how much information about a *-algebra can be gleaned from probabilistic properties of its generators. Some of the first results in this theme come from the theory of Voiculescu's free entropy: generators satisfying certain entropic assumptions generate von Neumann algebras which are non-Gamma, or prime, or do not admit Cartan subalgebras. Free Stein dimension -- a quantity I introduced with Nelson -- is a more recent quantity in a similar vein, which is robust under polynomial transformations and not trivial for variables which do not embeddable in R^\omega.

After giving a brief introduction to free probability and free entropy, I will speak on some recent improvements in related to free Stein dimension. We are now able to compute the free Stein dimension of direct sums, and of tensor products with finite dimensional algebras, which allow us to compute it in a large number of new examples. We are also able to give bounds on Stein dimension based on the presence of algebraic relations among generators. This is joint work with Brent Nelson.

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