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\(K\)-theory for groups acting on trees. (English) Zbl 0757.46059

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 979-986 (1991).
[For the entire collection see Zbl 0741.00020.]
In this expository article, the author gives a brief statement of various aspects of \(K\)-theory and \(KK\)-theory that carry over to the case of a group acting on an oriented tree, i.e., on a one-dimensional simply connected simplicial complex. First he mentions various approaches to \(K\)-theory for \(C^*\)-algebras and indicates how \(K\)-theory and \(KK\)- theory are used. Examples mentioned are the irrational rotation algebras and the reduced \(C^*\)-algebra of the free group with \(n\) generators. In the case of trees, the author mentions the theorem of G. G. Kasparov and G. Skandalis [C. R. Acad. Sci., Paris, Sér. I 310, No. 4, 171-174 (1990; Zbl 0705.19010)] that states \(\alpha\otimes_ \mathbb{C}\beta=1_{{\mathcal A}_ X}\) and \(\beta\otimes_{{\mathcal A}_ X}\alpha=\gamma^ \Gamma\) is idempotent where \({\mathcal A}_ X\) is the right analogue of the \(C^*\)-algebra of continuous functions on \(X\) when the simplicial complex \(X\) is a Bruhat-Tits building. Here \(\alpha\) and \(\beta\) are the Dirac and the dual Dirac elements in the \(KK_ n^ \Gamma({\mathcal A}_ X,\mathbb{C})\) and \(KK_ n^ \Gamma(\mathbb{C},{\mathcal A}_ X)\) respectively. He relates this theorem to the theorem of G. G. Kasparov [Funct. Anal. Appl. 7 (1973), 238-240 (1974; Zbl 0305.58017)] that states \(\alpha\otimes_ \mathbb{C}\beta=1_{C_ 0(X)}\) in \(KK_ 0^ G(C_ 0(X),C_ 0(X))\) and \(\beta\otimes_{C_ 0(X)}\alpha=\gamma^ G\) is an idempotent in \(KK_ 0^ G(\mathbb{C},\mathbb{C})\) where \(X=G/K\) with \(G\) a connected Lie group and \(K\) is its maximal compact subgroup.

MSC:

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