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

AbstractThis paper deals with a “naive” way of generalizing Kazhdan's property (T) to C∗-algebras. Our approach differs from the approach of Connes and Jones, which has already demonstrated its utility. Nevertheless, it turns out that our approach is applicable to a rather subtle question in the theory of C∗-Hilbert modules. Namely, we prove that a separable unital C∗-algebra A has property MI (module infinite—i.e. any countably generated self-dual Hilbert module over A is finitely generated and projective) if and only if A does not satisfy our definition of property (T). The commutative case was studied in an earlier paper