Model Theory and Dynamics of Classifiable C*-algebras

16.12.2022 15:00 - 15:45

Andrea Vaccaro (Université Paris Cité)




Abstract: One of the main contemporary themes of research in the study of operator algebras is the so-called Elliott Classification Program, a long-lasting project aiming to classify suitably regular C*-algebras (usually referred to as classifiable C*-algebras) by computable invariants arising from K-theory. After a brief introduction on C*-algebras and on this topic, I will present two branches of my research which both originate and are motivated by Elliott Classification Program.

The first one focuses on applications of continuous model theory in the study of classifiable C*-algebras. Elliott Classification Program concretely aims to reduce the study of the isomorphism relation between two classifiable C*-algebras to the isomorphism relation between the corresponding invariants, which in many cases of interest consist of discrete structures. I will show that an analogous reduction can be obtained, on approximately finite C*-algebras, for the relation of elementary equivalence, thanks to the employment of metric adaptations of Ehrenfeucht–Fraissé games and infinitary logic. As an application, I will sketch how this result allows to build families of approximately finite C*-algebras of arbitrarily high Scott rank.

The second segment of my talk will focus on amenable group actions on classifiable C*-algebras and on the crossed products arising from such dynamical systems. The crossed product is a classical C*-algebraic operation allowing to obtain a C*-algebra from a dynamical system on a topological space or on a C*-algebra. A problem which has recently attracted considerable attention is determining when the crossed product of a classifiable C*-algebra is again classifiable. I will give an overview of the state-of-the-art on this topic, focusing in particular on some recent results involving tools and ideas originating from the study of dynamical systems on topological spaces, such as almost finiteness and dynamical comparison.



Dekan R.I. Bot
SR 06, 1. OG, OMP