Foundations of an equivariant cohomology theory of Banach algebras. I
type: Item
instance of: scholarly article
title: Foundations of an equivariant cohomology theory of Banach algebras. I
zbMATH Open document ID: 0857.46050
author: Kjeld Knudsen Jensen
published in: Advances in Mathematics
publication date: 1996-05-19T00:00:00Z
review text: In several mathematical and physical situations the appearance of algebras is associated with special group actions. Of course, the introduction of an action can drastically change the equivariant cohomology from the classical ones and as those, its purpose is to help in describing and classifying these combined structures providing characterizations of certain classes [cf. \textit{D. Gong}, Mich. Math. J. 39, No. 3, 455-473 (1992; Zbl 0783.55005), \textit{S. Klimek}, \textit{W. Kondracki} and \textit{A. Lesniewski}, K-Theory 4, No. 3, 201-218 (1991; Zbl 0744.46063) and \textit{P. Baum} and \textit{A. Connes}, Lond. Math. Soc.Lect. Note Ser. 135, 1-20 (1988; Zbl 0685.46041)]. In this first part of a treatise, after an equivariant version of the basic definitions has been formulated in the purely algebraic and the Banach settings, special attention is paid to the dual equivariant modules. For a given (locally compact) topological group \(G\) acting continuously on a Banach (or \(C^*\)-)algebra \(\mathcal U\) and coherently on a Banach \({\mathcal U}\)-bimodule \(M\), the equivariant cohomology groups \(H^n_G({\mathcal U}, M)\) are then considered, in particular in lower dimensions, a dimension shift formula is also valid (with some restrictions), and sufficient conditions for the induced natural map \(\Phi: H^n_G({\mathcal U}, M)\to H^n({\mathcal U}, M)\) to be injective and/or surjective are given, providing an isomorphism, beyond the case of a trivial action, when \(G\) is finite with inner action on \(\mathcal U\). Next, the relative cohomology of \textit{M. Gerstenhaber} and \textit{S. D. Schack} [J. Pure Appl. Algebra 43, 53-74 (1986; Zbl 0603.16021)] is settled in the present context, and a chapter is devoted to cohomologically trivial algebras i.e., \(G\)-contractible and \(G\)-amenable. The theorems are as expected so far, modulo the necessary assumptions, which are stronger in the equivariant case. In fact, the ordinary notions still imply the equivariant on…
reviewed by: George F. Nassopoulos
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55N91
zbMATH DE Number: 859774
zbMATH Keywords: group actions
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: equivariant
MaRDI profile type: Publication
full work available at URL: aima.1996.0003
OpenAlex ID: W4210409554
DOI: 10.1006/AIMA.1996.0003
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description: scientific article; zbMATH DE number 859774