On groups with quasidiagonal C*-algebras

TL;DR

This paper investigates the quasidiagonality property of C*-algebras associated with discrete amenable groups, providing new characterizations, examples, and distinctions related to strong quasidiagonality.

Contribution

It offers a quantitative version of Rosenberg's theorem, characterizes quasidiagonality via embeddability, and explores examples including topological full groups and solvable groups.

Findings

Certain topological full groups have quasidiagonal C*-algebras.

Some amenable groups possess strongly quasidiagonal C*-algebras, while others do not.

The paper provides a new perspective on the relationship between group properties and C*-algebra quasidiagonality.

Abstract

We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of quasidiagonality for group C*-algebras in terms of embeddability of the groups. We consider several notable examples of groups, such as topological full groups associated with Cantor minimal systems and Abels' celebrated example of a finitely presented solvable group that is not residually finite, and show that they have quasidiagonal C*-algebras. Finally, we study strong quasidiagonality for group C*-algebras, exhibiting classes of amenable groups with and without strongly quasidiagonal C*-algebras.