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Une notion de nucléarité en K-théorie (d’après J. Cuntz). (A notion of nuclearity in K-theory (in the sense of J. Cuntz)). (French) Zbl 0653.46065

A notion of K-theoretic nuclearity is defined for \(C^*\)-algebras. For a given \(C^*\)-algebra A, it is given in terms of a certain A, A bimodule. It is proved that the Kasparov groups of K-theoretically nuclear \(C^*\)- algebras behave like those of nuclear algebras with respect to tensor products and exact sequences. Examples of non-K-theoretically nuclear \(C^*\)-algebras are given and analyzed. Specifically such examples are associated to connected simple Lie groups G of rank one, and discrete subgroups \(\Gamma\) \(\subset G\) with Kazhdan’s property T. Then, for such \(\Gamma\), \(C^*_{red}(\Gamma)\) is proved to be non-K-theoretically nuclear. As a corollary, it is noted that such \(C^*_{red}(\Gamma)\) cannot be K-equivalent to nuclear algebras.
Reviewer: P.E.T.Jörgensen

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
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