Speaker: Roland Speicher (Saarland University)
Date: June 8, 2022
Abstract: $q$-Gaussian random variables, for some fixed real $q$ with $-1\leq 1$, are of the form $a_i+a_i^*$, where the $a_i$ are operators satisfying the q-relations $ a_i a_j^* - qa_j^* a_i = \delta_{i j}$. Understanding the properties of the non-commutative distributions of those deformations of classical multivariate Gaussian distributions as well as their associated operator algebras -- in particular, whether and how they depend on $q$ -- has been of central interest in the last 30 years. I will give an introduction and survey on those q-relations and in particular report also some recent progress (from joint work with A.Miyagawa) on the existence of dual systems and conjugate systems for the $q$-Gaussians. Special focus is on the fact that those results are for the whole interval $(-1,+1)$, and not just for some restricted set of $q$.
