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

摘要

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.