Relative Yoneda cohomology for operator spaces
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.
Sz.-Nagy-Foias commutant lifting theorem
scientific article; zbMATH DE number 1211637
scientific article