Ideal Module Amenability of Triangular Banach Algebras

Abstract

‎Let $A$ and $B$ be unital Banach algebras and $M$ be an unital Banach $A‎, ‎B$-module‎. ‎In this paper we define the concept of the $(n)$-ideal module amenability of Banach algebras and investigate the relation between the $(2n-1)$-ideal module amenability of triangular Banach algebra‎ ‎$\mathcal T=\left[\begin{array}{rr} A & M \\ & B \end{array} \right]$ (as a $‎ ‎\mathfrak{T}=\set{\Mat{\alpha}{}{\alpha}; \alpha\in \mathfrak{A}}‎ ‎$-module) and $(2n-1)$-ideal module amenability of $A$ and $B$ (as an $\mathfrak{A}$-module)‎. ‎Finally‎, ‎in the case that $A=B=M=\ell^1(S)$‎, ‎for unital and commutative inverse semigroup $S$ with idempotent set $E$‎, ‎we show that‎ ‎%$\left[\begin{array}{rr} \ell^1(S) & \ell^1(S) \\ & \ell^1(S)\end{array} \right]$‎ ‎$\mathcal T$ as a $\mathfrak{T}$-module is $(2n-1)$-ideal module amenable while is not module amenable‎.