n-Weak Module Amenability of Triangular Banach Algebras

2013/01/15 by Abasalt Bodaghi, Bodaghi, Abasalt, Ali Jabbari +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1301.3237

arxiv created 2013/01/15 · arxiv updated 2013/01/16

Abstract

Let \mathcal A, \mathcal B be Banach \mathfrak A-modules with compatible actions and \mathcal M be a left Banach \mathcal A-\mathfrak A-module and a right Banach \mathcal B-\mathfrak A-module. In the current paper, we study module amenability, n-weak module amenability and module Arens regularity of the triangular Banach algebra \mathcal T=[ cc \mathcal A & \mathcal M & \mathcal B ] (as an \mathfrak T:=[ cc α& & α ] | α∈\mathfrak A-module). We employ these results to prove that for an inverse semigroup S with subsemigroup E of idempotents, the triangular Banach algebra \mathcal T₀=[ cc ℓ¹(S)& ℓ¹(S) & ℓ¹(S) ] is permanently weakly module amenable (as an \mathfrak T₀=[ cc ℓ¹(E)& & ℓ¹(E) ]-module). As an example, we show that \mathcal T₀ is \mathfrak T₀-module Arens regular if and only if the maximal group homomorphic image G_S of S is finite.

Related

• Module Amenability for Semigroup Algebras
• Module approximate amenability of Banach algebras
• Module super-amenability for semigroup algebras
• Module Pseudo-amenability of Banach algebras
• Arens regularity of module actions and weak amenability of Banach algebras