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A \(C^{*}\)-analogue of Kazhdan’s property (T). (English) Zbl 1130.46036

There are two essential approaches to the definition of Kazhdan’s property (T) (cf.D.A.Kazhdan [Funct.Anal.Appl.1, 63–65 (1967; Zbl 0168.27602); translation from Funkts.Anal.Prilozh.1, No. 1, 71–74 (1967)]) for topological groups. The first one uses the concept of \(\varepsilon\)-invariant vector and the second applies Fell’s topology. Using the notion of correspondence an analogue of property (T) for \(W^*\)-algebras was introduced by A.Connes and V.Jones [Bull.Lond.Math.Soc.17, 57–62 (1985; Zbl 1190.46047)] in which they proved that a countable discrete group \(G\), whose von Neumann algebra \(L(G)\) is factorial, has Kazhdan’s property if and only if \(L(G)\) has property (T).
In the paper under review, the authors give the following naive generalization of property (T) in the context of \(C^*\)-algebras: A unital \(C^*\)-algebra \(A\) has property (TP) (property (T) at some point of its spectrum) if there exists an isolated point with respect to Fell’s topology in the unitary dual \(\widehat{A}\) such that the corresponding representation is finite-dimensional. Following E.V.Troitsky [Proc.Am.Math.Soc.131, No. 11, 3411–3422 (2003; Zbl 1113.46058)], a unital \(C^*\)-algebra \(A\) is called MI (module-infinite) if each self-dual countably generated Hilbert \(A\)-module is finitely generated and projective. The main theorem of the paper reads as follows: Suppose that \(A\) is a separable unital \(C^*\)-algebra. Then \(A\) is MI if and only if \(\widehat{A}\) does not have property (TP). For commutative algebras, this result was proved by E.V.Troitsky [loc.cit.].

MSC:

46L08 \(C^*\)-modules
46L05 General theory of \(C^*\)-algebras
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References:

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