Cohomological characterizations of biprojective and biflat Banach algebras
type: Item
instance of: scholarly article
title: Cohomological characterizations of biprojective and biflat Banach algebras
zbMATH Open document ID: 0951.46042
DOI: 10.1007/PL00010082
author: Yurií V. Selivanov
published in: Monatshefte für Mathematik
publication date: 2000-12-18T00:00:00Z
review text: The cohomology of a Banach algebra \(A\) is here treated as a particular case of the Ext functor for bimodules, by employing the so-called entwining resolution and arguments of \textit{A. Ya. Khelemskij} [``The homology of Banach and topological algebras'', Moskva (1986; Zbl 0608.46046)]. In fact, this is an \(A\)-bimodule resolution that is projective if \(A\) is biprojective and, furthermore, flat if \(A\) is biflat. By computing all the (continuous Hochschild) cohomology groups \(H^n(A, X)\) and \(H^n(A,X^*)\) respectively, where \(X^*\) stands for the Banach \(A\)-bimodule dual to \(X\), some cohomological characterizations of the biprojectivity and of the biflatness for Banach algebras are obtained. In conclusion, \(A\) is biprojective (resp. biflat) if and only if \(H^1(A,{\mathcal M}(X))= 0\) for each (resp. each dual) Banach \(A\)-bimodule \(X\), \({\mathcal M}(X)\) denoting the bimodule of double multipliers of \(X\).
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 18G15
zbMATH DE Number: 1340543
zbMATH Keywords: continuous Hochschild cohomology groups
zbMATH Keywords: Ext functor
zbMATH Keywords: bimodules
zbMATH Keywords: entwining resolution
zbMATH Keywords: biprojective
zbMATH Keywords: biflat
zbMATH Keywords: double multipliers
reviewed by: George F. Nassopoulos
MaRDI profile type: Publication
full work available at URL: pl00010082
OpenAlex ID: W1989723108
description: scientific article; zbMATH DE number 1340543