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

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 Popaʼs 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.