n-Weak Module Amenability of Triangular Banach Algebras

Abstract

Let A, B be Banach A-modules with compatible actions and M be a left Banach A-A-module and a right Banach B-A-module. In the current paper, we study module amenability, n-weak module amenability and module Arens regularity of the triangular Banach algebra T = A M B (as an T :=. We employ these results to prove that for an inverse semigroup S with subsemigroup E of idempotents, the triangular Banachis permanently weakly module amenable (as an T 0 = 1 (E) 1 (E) -module). As an example, we show that T 0 is T 0 -module Arens regular if and only if the maximal group homomorphic image G S of S is finite.

Pseudo-amenability and pseudo-contractibility of restricted semigroup algebra

“…Recently, the notions of module pseudo-amenability and module pseudo-contractibility in Banach algebras were introduced in [2], where necessary and sufficient conditions were particularly obtained for the semigroup algebra l 1 (S) and its dual to be l 1 (E)-module pseudo-amenable for every inverse semigroup S with subsemigroup E of idempotent.…”

Weak amenability for the second dual of Banach modules

“…He showed that the group algebra L 1 .G/ is amenable if and only if G as a group is amenable. Subsequently, various generalizations of this notion have been studied by a number of authors (for instance, see [2,6,11,19]). In other words, all authors investigated and obtained some results related to amenability of Banach algebras while many Banach algebras can be considered as a Banach module over another Banach algebras.…”