Haagerup property for C*-algebras and rigidity of C*-algebras with property (T)

TL;DR

This paper explores the Haagerup property for C*-algebras, providing new examples and establishing permanence results, while also examining rigidity properties of C*-algebras with property (T) and their implications for group C*-algebras.

Contribution

It introduces new examples of C*-algebras with the Haagerup property, proves their permanence, and investigates rigidity phenomena for C*-algebras with property (T).

Findings

Nuclear C*-algebras with faithful traces have the Haagerup property.

The class of C*-algebras with the Haagerup property is large.

Reduced group C*-algebras of certain groups exhibit rigidity properties.

Abstract

We study the Haagerup property for C*-algebras. We first give new examples of C*-algebras with the Haagerup property. A nuclear C*-algebra with a faithful tracial state always has the Haagerup property, and the permanence of the Haagerup property for C*-algebras is established. As a consequence, the class of all C*-algebras with the Haagerup property turns out to be quite large. We then apply Popa's results and show the C*-algebras with property (T) have a certain rigidity property. Unlike the case of von Neumann algebras, for the reduced group C*-algebras of groups with relative property (T), the rigidity property strongly fails in general. Nevertheless, for some groups without nontrivial property (T) subgroups, we show a rigidity property in some cases. As examples, we prove the reduced group C*-algebras of the (non-amenable) affine groups of the affine planes have a rigidity property.