Equivariant K-theory and freeness of group actions on \(C^ *\)-algebras
In K-theory for \(C^*\)-algebras it is natural to take concepts from topological K-theory for topological spaces and to see what they give in the general ``non-commutative'' context. The author does this here with the concept of a K-free action of a compact group G on a compact space X, which was introduced by Atiyah and Segal and shown to be equivalent to freeness of the action of G. Let I(G) be the ideal of elements of dimension 0 in the representation ring R(G) of G. Then an action of G on X is K-free if there exists n such that \(I(G)^ nK^*_ G(X)=0\), where \(K^*_ G(X)\) is the equivariant K-theory of X. Accordingly, an action of G on a \(C^*\)-algebra is defined to be K-free if there is n such that \(I(G)^ nK^ G_*(A)=0.\) Freeness of a G-action on a \(C^*\)-algebra is not a well defined concept. Thus the author investigates if K-freeness could be used to replace it and also compares it to other substitutes of freeness for G- actions. Possible substitutes are e.g. fullness of the strong Connes spectrum, KK-freeness or also K-saturation: the action is K-saturated if the natural map \(K_*(A^ G)\to K^ G_*(A)\), where \(A^ G\) is the fixed point algebra, is an isomorphism. The result of the investigation is that each of the possible substitutes has its drawbacks and that the connections between them are rather messy except for the case of the finite cyclic group or \(S^ 1\) acting on a separable type I algebra where at least part of them coincide and are equivalent to free action on Prim(A). The book contains a good and detailed introduction to equivariant K- theory and KK-theory, many examples of actions on \(C^*\)-algebras and their K-theory and a wealth of further related material. These points make for its real value.
scientific article; zbMATH DE number 4028180