K-theory for certain group \(C^*\)-algebras. (English) Zbl 0542.46031
The use of K-theoretic techniques in \(C^*\)-algebras has led to the solution of several outstanding problems, among them the conjecture of R. V. Kadison that \(C^*_ r(F_ n)\), the reduced \(C^*\)-algebra of the free group on n generators (\(n\geq 2)\), has no nontrivial projections. This was resolved affirmatively by M. V. Pimsner and D. Voiculescu [J. Oper. Theory 8, 131-156 (1982)] as a corollary to a remarkable theorem which describes the K-groups for any reduce crossed product of a \(C^*\)-algebra by an action of a free group.
This paper originated in the author’s attempt to understand the work of Pimsner and Voiculescu, and in particular to see whether their methods could be used to give a simpler proof than \(K_ 0(C^*_ r(F_ n))=Z.\) By slightly adapting their approach, we are able to give a description of \(K_*(C^*_ r(\Gamma)\) for any group \(\Gamma\) which is a free product of countable amenable groups. Since this work was done there have been further significant developments in this area. J. Cuntz [J. Reine Angew. Math. 344, 180-195 (1983; Zbl 0511.46066)] has used the machinery of Kasparov’s KK-theory to give a neat computation of the K-groups for the reduced \(C^*\)-algebras of a class of groups apparently more general than that considered in this paper. In two further papers [Lecture Notes in Math. 1031, 31-45 (1983); “K-theory and \(C^*\)-algebras”, to appear in Proc. of K-theory Conf. Bielefeld 1982] he has shown how KK-theory can be developed using what in this paper are called difference maps as the basic elements of the theory. In fact, the difference maps constructed in this paper furnish some instructive examples of elements of certain KK-groups and may be usefully contemplated by anyone wishing to learn KK-theory.
This paper originated in the author’s attempt to understand the work of Pimsner and Voiculescu, and in particular to see whether their methods could be used to give a simpler proof than \(K_ 0(C^*_ r(F_ n))=Z.\) By slightly adapting their approach, we are able to give a description of \(K_*(C^*_ r(\Gamma)\) for any group \(\Gamma\) which is a free product of countable amenable groups. Since this work was done there have been further significant developments in this area. J. Cuntz [J. Reine Angew. Math. 344, 180-195 (1983; Zbl 0511.46066)] has used the machinery of Kasparov’s KK-theory to give a neat computation of the K-groups for the reduced \(C^*\)-algebras of a class of groups apparently more general than that considered in this paper. In two further papers [Lecture Notes in Math. 1031, 31-45 (1983); “K-theory and \(C^*\)-algebras”, to appear in Proc. of K-theory Conf. Bielefeld 1982] he has shown how KK-theory can be developed using what in this paper are called difference maps as the basic elements of the theory. In fact, the difference maps constructed in this paper furnish some instructive examples of elements of certain KK-groups and may be usefully contemplated by anyone wishing to learn KK-theory.
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 46L55 | Noncommutative dynamical systems |
Citations:
Zbl 0511.46066References:
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