The Atiyah-Segal completion theorem for \(C^*\)-algebras
type: Item
instance of: scholarly article
title: The Atiyah-Segal completion theorem for \(C^*\)-algebras
zbMATH Open document ID: 0709.46034
DOI: 10.1007/BF00534138
author: N. Christopher Phillips
published in: \(K\)-Theory
publication date: 1989-01-01T00:00:00Z
review text: 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.
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 4165732
zbMATH Keywords: crossed product of C*-algebras
zbMATH Keywords: equivariant K-theory
zbMATH Keywords: representative K- theory group
zbMATH Keywords: stable outer multiplier algebra
zbMATH Keywords: homotopy equivariant
zbMATH Keywords: long exact sequences
zbMATH Keywords: Atiyah-Segal completion theorem
zbMATH Keywords: compact Lie group
zbMATH Keywords: representable K-theory for sigma-C*-algebras
zbMATH Keywords: classifying space
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00534138
OpenAlex ID: W1976729017
description: scientific article; zbMATH DE number 4165732