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K-groupes of reduced crossed products by free groups. (English) Zbl 0533.46045

Let \(F_ n\) be a free group of n generators. Let A be a \(C^*\)-algebra on which \(F_ n\) is acting. The authors consider the reduced crossed product of A by \(F_ n\), \(A\times_{\alpha r}F_ n\). Using a six terms cyclic exact sequence they compute the K-groups of \(A\times_{\alpha r}F_ n\). In particular they show that for the reduced \(C^*\)-algebra of \(F_ n\), \(C^*\!_ r(F_ n)\), the following holds: \(K_ 0(C^*\!_ r(F_ n))=Z\) and \(K_ 1(C^*\!_ r(F_ n))=Z^ n.\)
This way they infer that there are no projections but 0 and 1 in the reduced \(C^*\)-algebra of a free group, solving a problem of R. V. Kadison.
Together with results of J. M. Cohen, J. Funct. Anal. 33, 211-216 (1979; Zbl 0424.46043) and J. Cuntz, Proc. Symp. Pure Math. 38, Part 1, Kingston/Ont. 1980, 81-84 (1982; Zbl 0502.46050) this shows that the full group \(C^*\)-algebra and the reduced \(C^*\)-algebra of \(F_ n\) have the same \(K_ 0\) and \(K_ 1\) groups, i.e. the same K-theory.
Reviewer: P.Kruszyński

MSC:

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46L55 Noncommutative dynamical systems
46L05 General theory of \(C^*\)-algebras
46L40 Automorphisms of selfadjoint operator algebras
55N15 Topological \(K\)-theory
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