A C*-analogue of Kazhdan's property (T)

Abstract

This paper deals with a "naive" way of generalization of the Kazhdan's property (T) to C*-algebras. This approach differs from the approach of Connes and Jones, which has already demonstrated its utility. Nevertheless it turned out that our approach is applicable to the following rather subtle question in the theory of C*-Hilbert modules. We prove that a separable unital C*-algebra A has property MI (module-infinite, i.e., any C*-Hilbert module over A is self-dual if and only if it is finitely generated projective) if and only if it has not property (T1P) (property (T) at one point}, i.e. there exists in the unitary dual of A a finite dimensional isolated point. The commutative case was studied in a previous paper.