Topology on \(S^{-1}S\) for Banach algebras
type: Item
instance of: scholarly article
title: Topology on \(S^{-1}S\) for Banach algebras
zbMATH Open document ID: 0866.55018
DOI: 10.1016/0040-9383(95)00050-X
author: Michael Paluch
published in: Topology
publication date: 1997-07-20T00:00:00Z
review text: Let \(\mathcal P\) be the category of finitely generated projective modules over a Banach algebra with identity \(\Lambda\), and \(\Lambda(\Delta^p)\) the ring of continuous \(\Lambda\)-valued functions on \(\Delta^p\). Quillen's \(S^{-1}S\)-construction provides a model for the algebraic \(K\)-theory of \(\Lambda\) [\textit{D. Grayson}, Lect. Notes Math. 551, 217-240 (1976; Zbl 0362.18015)]. This note shows that there is a topological enrichment \(S^{-1}S^{\text{top}}({\mathcal P})\) of the category \(S^{-1}S({\mathcal P})\) which computes the topological \(K\)-theory of \(\Lambda\): \[ K^{\text{top}}_p(\Lambda)\cong \pi_p S^{-1} S^{\text{top}}({\mathcal P}). \] The obvious functor \(S^{-1}S({\mathcal P})\to S^{-1} S^{\text{top}}({\mathcal P})\) induces the expected map from the algebraic to the topological \(K\)-groups of \(\Lambda\). Besides, this construction gives rise to a spectral sequence \(E^1_{p,q}(\Lambda)=K^{\text{alg}}_q \Lambda(\Delta^p)\) that converges to \(K^{\text{top}}_{p+q}(\Lambda)\).
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 19D50
zbMATH DE Number: 929554
zbMATH Keywords: category
zbMATH Keywords: projective modules
zbMATH Keywords: Banach algebra
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: topological \(K\)-theory
reviewed by: Frédéric Patras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 929554