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

• Published in: Advances in mathematics (New York. 1965) Vol. 216; no. 1; pp. 75 - 88
• Main Authors: Pavlov, A.A., Troitsky, E.V.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2007
• Subjects: Dual object; Hilbert [formula omitted]-module; Kazhdan's property (T); Kazhdan's property (T); 46L; Hilbert C ∗ -module; Dual object
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2007.05.003

This 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.