The first module (σ, τ)-cohomology group of triangular Banach algebras of order three

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson's amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243-254]. The weak module amenability of the triangular Banach algebra T-2 = [GRAPHICS] , where A and B are Banach algebras (with U-module structure) and M is a Banach A, B-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949-958], and they showed that the weak module amenability of 2 x 2 triangular Banach algebra T-2 (as an I := { [GRAPHICS] vertical bar alpha is an element of U}- bimodule) is equivalent with the weak module amenability of the corner algebras A and B (as Banach U-bimodules). The main aim of this paper is to investigate the module (sigma, tau)-amenability and weak module (sigma, tau) -amenability of the triangular Banach algebra T of order three, where sigma and tau are U-module morphisms on T. Also, we give some results for semigroup algebras.