Module amenability of the second dual and module topological center of semigroup algebras

Abstract

Let S be an inverse semigroup with an upward directed set of idempotents E. In this paper we define the module topological center of second dual of a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find it for ℓ 1 (S) * * (as an ℓ 1 (E)-module). We also prove that ℓ 1 (S) * * is ℓ 1 (E)-module amenable if and only if an appropriate group homomorphic image of S is finite.2000 Mathematics Subject Classification. 46H25.

A generalization of the $$n$$ n -weak module amenability of banach algebras

“…It is shown in [4] that C/ ≈ is isomorphic to Z for an equivalence relation ≈ defined on C, and hence l 1 (C/ ≈) is amenable. By [4, Proposition 3.3], l 1 (C) is l 1 (E C )-module amenable.…”

“…Then the quotient S/ ≈ is a discrete group (see [4,20]), and it is isomorphic to the maximal group homomorphic image G S of S (see [19]). As in [21,Theorem 3.3], we may observe that l 1 (S)/J ∼ = l 1 (S/ ≈).…”

Module character inner amenability of Banach algebras

“…We consider an equivalence relation on S as follows s % t , d s À d t 2 J l 1 ðSÞ ðs; t 2 SÞ: For inverse semigroup S, the quotient semigroup S= % is discrete group and so l 1 ðS= %Þ has an identity (see [2,13]). Indeed, S= % is homomorphic to the maximal group homomorphic image G S of S (see [11,14]).…”

n-Weak Module Amenability of Triangular Banach Algebras

“…In general case, J ·(A ⊗A) is not a subset of I and thus (A ⊗A)/I is not always an A/J -module. We say the Banach algebra A acts trivially on A from left (right) if there is a continuous linear functional f on A such that α · a = f (α)a (a · α = f (α)a), for each α ∈ A and a ∈ A (see also [2]). The following lemma is proved in [7, Lemma 3.13].…”