Pimsner, M.; Voiculescu, D. K-groupes of reduced crossed products by free groups. (English) Zbl 0533.46045 J. Oper. Theory 8, 131-156 (1982). 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 Cited in 3 ReviewsCited in 68 Documents 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 Keywords:K-theory for \(C^*\)-algebras; reduced crossed product; six terms cyclic exact sequence; K-groups Citations:Zbl 0424.46043; Zbl 0502.46050 PDFBibTeX XMLCite \textit{M. Pimsner} and \textit{D. Voiculescu}, J. Oper. Theory 8, 131--156 (1982; Zbl 0533.46045)