The first cohomology group of module extension Banach algebras

abstract. Let A be a Banach algebra and X be a Banach A-bimodule. Then S = A⊕X, the l1-direct sum of A and X becomes a module extension Banach algebra when equipped with the algebra product (a, x).(a′, x′) = (aa′, ax′ + xa′). In this paper we compute the first cohomology group H1(S,S) for module extension Banach algebras S. Also we obtain results on n-weak amenability of commutative module extension Banach algebras. We have shown that there are many different examples of non-n-weak amenable Banach algebras.