Relative Yoneda cohomology for operator spaces
type: Item
instance of: scholarly article
title: Relative Yoneda cohomology for operator spaces
zbMATH Open document ID: 0917.46052
author: Vern I. Paulsen
published in: Journal of Functional Analysis
publication date: 1999-07-19T00:00:00Z
review text: Versions of Yoneda cohomology [cf. \textit{M. Gerstenhaber} and \textit{S. D. Schack}, NATO ASI Ser., Ser. C 247, 11-264 (1988; Zbl 0676.16022)] are constructed for operator spaces \(X\in {_A}O_B\) which are the \(A\)-\(B\) bimodules over (unital) operator algebras \(A,B\) with completely bounded (c.b.) module pairings \(A\times X\times B\to X\). An accent is made on the relative theory treating the groups of equivalence classes of suitable (\(C\)-\(D\)-split with respect to the unital subalgebras \(C\subset A\), \(D\subset B\)) c.b. \(n\)-extensions of \(X\) by \(Y\in {_A}O_B\). Due to the derived functor realization, this theory is isomorphic with the relevant c.b. Hochschild cohomology. The concepts of relative injectivity, projectivity and amenability are studied in this setting. A remarkable result is that every von Neumann algebra \(M\) is relatively injective as a bimodule over itself, which entails the property \(H_{\text{cb}}^n(M;M)=0\) established by E. Christensen and A. M. Sinclair. A number of results are obtained which have counterparts in the purely algebraic or Banach algebra/Banach module setting [cf. \textit{A. Ya. Khelemskij}, ``The homology of Banach and topological algebras'' (1989; Zbl 0695.46033)]. Note also a homological generalization of the Sz.-Nagy-Foias commutant lifting theorem.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 18E25
zbMATH DE Number: 1211637
zbMATH Keywords: operator space
zbMATH Keywords: completely bounded mapping
zbMATH Keywords: bimodule
zbMATH Keywords: relative Yoneda cohomology
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: relative injectivity
zbMATH Keywords: projectivity
zbMATH Keywords: amenability
zbMATH Keywords: derived functor realization
zbMATH Keywords: Sz.-Nagy-Foias commutant lifting theorem
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
full work available at URL: jfan.1997.3208
OpenAlex ID: W2056052803
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DOI: 10.1006/JFAN.1997.3208
description: scientific article; zbMATH DE number 1211637