  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.
