Cohomological characterizations of biprojective and biflat Banach algebras
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 \(X\).
scientific article; zbMATH DE number 1340543