The Atiyah-Segal completion theorem for \(C^*\)-algebras
The author, in the previous paper [Representable K-theory for \(\sigma\)- \(C^*\)-algebras, k-theory 3, 444-478 (1989; review above)] has defined his representative K-theory group \(RK_ 0(A)\) as the group \(UQ(A)/U_ 0Q(A)\) of path components of the unitary group of the stable outer multiplier algebra \(Q(A)=M({\mathcal K}\otimes A)/({\mathcal K}\otimes A)\) and then \(RK_ i(A)\) is defined as \(RK_ 0\) of the ith iterated suspension \(S^ iA:=A\otimes {\mathbb{C}}_ 0({\mathbb{R}})\otimes...\otimes {\mathbb{C}}_ 0({\mathbb{R}}).\) This representable K-theory admits a majority of important properties, namely, the homotopy equivariant, long exact sequences etc... With this theory the author, in the paper under review generalizes the Atiyah-Segal completion theorem to \(C^*\)-algebras as follows. Let A be a \(C^*\)-algebra with a continuous action of the compact Lie group G. If \(K^ G_*(A)\) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion \(K^ G_*(A){\hat{\;}}\) is isomorphic to \(RK_*([A\otimes C(EG)]^ G)\), where \(RK_*\) is representable K-theory for \(\sigma\)-C\({}^*\)-algebras and EG is a classifying space for G. As a corollary, the author shows that if \(\alpha\) and \(\beta\) are homotopic actions of G, and if \(K_*(C^*(G,A,\alpha))\) and \(K_*(C^*(G,A,\beta))\) are finitely generated, then \(K_*(C^*(G,A,\alpha)){\hat{\;}}\cong K_*(C^*(G,A,\beta)){\hat{\;}}.\) The author gives examples to show that this isomorphism fails without the completions. However, he proves that this isomorphism does hold without the completion if the homotopy is requird to be norm continuous.
representable K-theory for sigma-C*-algebras
scientific article; zbMATH DE number 4165732
scientific article