The first cohomology group of module extension Banach algebras (Q640277)

The first cohomology group of a Banach algebra \(A\) with coefficients in a Banach \(A\)-bimodule \(X\), denoted by \(H^1(A,X)\), is the quotient space of the space of all continuous derivations from \(A\) to \(X\) modulo the space of inner derivations. An algebra \(A\) is called \(n\)-weakly amenable if \(H^1(A, A^{(n)})= \{0\}\), where \(A^{(n)}\) is the \(n\)th conjugate space (module) of \(A\). It is often important to calculate the first cohomology group for different types of Banach algebras; for example, \textit{B.\,E. Forrest} and \textit{L.\,W. Marcoux} [Trans. Am. Math. Soc. 354, No. 4, 1435--1452 (2002; Zbl 1014.46017)] and [Indiana Univ. Math. J. 45, No. 2, 441--462 (1996; Zbl 0890.46035)] have determined the first cohomology group of a class of Banach algebras which they called triangular Banach algebras. Also, \textit{Y. Zhang} [Trans. Am. Math. Soc. 354, No. 10, 4131--4151 (2002; Zbl 1008.46019)] has proved several general results on the \(n\)-weak amenability of module extension Banach algebras. Note that module extension Banach algebras are generalized forms of triangular Banach algebras. Based upon ideas from [Zbl 0890.46035], the authors of the paper under review compute the first cohomology group \(H^1({ A} \oplus X,{ A} \oplus X)\) for the module extension Banach algebra \({ A} \oplus X\). In particular, they calculate the first cohomology group \(H^1({ A} \oplus X,Y)\) for an \(A\)-bimodule \(Y\) which is an \({ A} \oplus X\)-bimodule in a canonical fashion. Finally, the authors illustrate the scope of their results by some interesting examples yielding some non-\(n\)-weakly amenable Banach algebras.