Representable K-theory for \(\sigma\)-C\({}^*\)-algebras
type: Item
instance of: scholarly article
title: Representable K-theory for \(\sigma\)-C\({}^*\)-algebras
zbMATH Open document ID: 0709.46033
DOI: 10.1007/BF00534137
author: N. Christopher Phillips
published in: \(K\)-Theory
publication date: 1989-01-01T00:00:00Z
review text: In the category of \(\sigma\)-C\({}^*\)-algebras (i.e. the countable inverse limits of \(C^*\)-algebras) the author defines his ``representative'' K-theory groups \(RK_ 0(A)\) as the group \(UQ(A)/U_ 0Q(A)\) of path components of the unitary group of the stable outer multiplier algebra \(Q(A)=M({\mathcal K}\otimes A)/({\mathcal K}\otimes A)\), where \({\mathcal K}\) is the algebra of compact operators on a separable infinite- dimensional Hilbert space. And naturally the \(RK_ i(A)\) groups are defined as \(RK_ 0\) of the ith iterated suspension \(S^ iA:=A\otimes C_ 0({\mathbb{R}})\otimes...\otimes C_ 0({\mathbb{R}}).\) The obtained theory is homotopy invariant, has long exact sequences and a Milnor \(\lim^ 1_{\leftarrow}\)-sequence, and satisfies Bott periodicity. On \(C^*\)- algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X.
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 4165731
zbMATH Keywords: category of \(\sigma \)-C\({}^ *\)-algebras
zbMATH Keywords: countable inverse limits of \(C^ *\)-algebras
zbMATH Keywords: stable outer multiplier algebra
zbMATH Keywords: homotopy invariant
zbMATH Keywords: long exact sequences
zbMATH Keywords: Milnor \(\lim ^ 1_{\leftarrow }\)-sequence
zbMATH Keywords: Bott periodicity
zbMATH Keywords: countable direct limit X of compact Hausdorff spaces
zbMATH Keywords: representable K-theory
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00534137
OpenAlex ID: W2048611604
description: scientific article; zbMATH DE number 4165731