Mario Klisse: On the isomorphism class of q-Gaussian C*-algebras

Abstract

In 1991 Bozejko and Speicher introduced a non-commutative version of Brownian motion by defining a family of algebras depending on a parameter \(−1 \leq q \leq 1\) that are nowadays commonly known as the q-Gaussian algebras. These algebras interpolate between the extreme Bosonic case q = 1 and the Fermionic case q = −1. For q = 0 they coincide with Voiculescu’s free Gaussians. The q-Gaussians can be studied on the level of *-algebras, on the level of C*-algebras, and on the level of von Neumann algebras. Whereas it is easily seen that in the *-algebraic setting the q-Gaussians all coincide, as soon as one passes to the operator algebraic level the question for the dependence on the parameter q becomes notoriously difficult.

After introducing the necessary background on q-Gaussians, by considering the so-called Akemann-Ostrand property of the canonical inclusion we will discuss the dependence of the isomorphism class of q-Gaussian C*-algebras on the parameter q. This partially answers a question by Nelson and Zeng.

The talk is biased on joint work with Matthijs Borst, Martijn Caspers and Mateusz Wasilewski.