On Weak Generalized Amenability of Triangular Banach Algebras

Abstract

Let A1, A2 be unital Banach algebras and X be an A1 − A2− module. Applying the concept of module maps, (inner) module generalized derivations and generalized first cohomology groups, we present several results concerning the relations between module generalized derivations from Ai into the dual space A∗ i (for i = 1, 2) and
such derivations from the triangular Banach algebra of the form T := A1 X 0 A2 into the associated triangular T − bimodule T ∗ of the form T ∗ :=
A∗ 1 X∗ 0 A∗ 2 . In particular, we show that the so-called generalized first cohomology group from T to T ∗ is isomorphic to the directed sum of the generalized first cohomology group from A1 to A∗ 1 and the generalized first cohomology group from A2 to A∗ 2. Finally, Inspiring the above concepts, we establish a one to one corresponding between weak (resp. ideal) generalized amenability of T and those amenability of Ai (i = 1, 2).