Foundations of an equivariant cohomology theory for Banach algebras. II
type: Item
instance of: scholarly article
title: Foundations of an equivariant cohomology theory for Banach algebras. II
zbMATH Open document ID: 0955.46043
author: Kjeld Knudsen Jensen
published in: Advances in Mathematics
publication date: 2001-02-21T00:00:00Z
review text: The author continues the investigation of the equivariant cohomology theory for Banach algebras [for part I see ibid. 117, No. 1, 52-146 (1996; Zbl 0857.46050)]. In this paper the author studies the relation between the equivariant cohomology groups and the standard cohomology groups of the crossed product algebra. This paper is organized in three chapters numbered IV, V, and VI. In Chapter IV, the construction of the covariance algebra as exhibited in the paper [\textit{S. Doplicher}, \textit{D. Kastler} and \textit{D. W. Robinson}, Commun. Math. Phys. 3, 1-28 (1966; Zbl 0152.23803)], is given. The same construction applies to a Banach covariance module, i.e. a Banach module over the covariant algebra. When \(G\) is a locally compact group acting on \({\mathfrak A}\), the notation \({\mathfrak A}^G_1\) of the previous paper for the covariance algebra is used; the covariance module constructed from a Banach \(G\)-\({\mathfrak A}\)-module \({\mathcal M}\) is denoted by \({\mathcal M}^G_1\). An action of the group \(G\) on the covariance algebra \({\mathfrak A}^G_1\) and the covariance module \({\mathcal M}^G_1\) is introduced. It is proved that with these action \({\mathcal M}^G_1\) is an equivariant \({\mathfrak A}^G_1\)-module. The author considers the question when a covariance algebra is amenable. In Chapter V, the relation between equivariant cohomology and various cohomology theories of the covariance algebra with coefficients in the covariance module is considered. In Chapter VI, there are considered the equivariant cohomology theories defined in the papers [\textit{S. Klimek}, \textit{W. Kondracki} and \textit{A. Lesniewski}, K-Theory 4, No. 3, 201-218 (1991; Zbl 0744.46063); \textit{D. Gong}, Mich. Math. J. 39, No. 3, 455-473 (1992; Zbl 0783.55005)]. It is proved that if the group \(G\) is finite or the algebra is finite-dimensional, the equivariant theories of the previous papers are special cases of the theory developed in part I of the author's paper.
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 1417608
zbMATH Keywords: equivariant cohomology theory
zbMATH Keywords: equivariant cohomology groups
zbMATH Keywords: standard cohomology groups
zbMATH Keywords: crossed product algebra
zbMATH Keywords: covariance algebra
zbMATH Keywords: Banach covariance module
MaRDI profile type: Publication
full work available at URL: aima.1999.1838
OpenAlex ID: W2045124584
cites work: Q5509066
cites work: On the vanishing of \(H^ n(\mathcal A,\mathcal A^*)\) for certain \(C^*\)- algebras
cites work: Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\)
cites work: On the cohomology of operator algebras
cites work: Covariance algebras in field theory and statistical mechanics
cites work: Excision of equivariant cyclic cohomology of topological algebras
cites work: The local structure of twisted covariance algebras
cites work: All nuclear C*-algebras are amenable
cites work: Q4692966
cites work: Foundations of an equivariant cohomology theory of Banach algebras. I
cites work: Approximate Diagonals and Cohomology of Certain Annihilator Banach Algebras
cites work: Cohomology of operator algebras. III : reduction to normal cohomology
cites work: Q3910959
cites work: Q4359479
cites work: Cohomology of operator algebras. I: Type I von Neumann algebras
cites work: Cohomology of operator algebras. II: Extended cobounding and the hyperfinite case
cites work: Equivariant entire cyclic cohomology. I: Finite groups
cites work: Duality Theory for Covariant Systems
cites work: Amenability for twisted covariance algebras and group \(C^*\)-algebras
cites work: Unitary representations of group extensions. I
cites work: Q5332447
cites work: Q3205815
cites work: Q5644619
cites work: Amenability of crossed products of \(C^*\)-algebras
cites work: Q5667036
cites work: Q4405392
cites work: Produits croises d'une \(C^ *\)-algèbre par un groupe d'automorphismes
DOI: 10.1006/AIMA.1999.1838
description: scientific article; zbMATH DE number 1417608