Symmetric module and connes amenability

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 - . We employ these results to prove that for an inverse semigroup S with subsemigroup E of idempotents, the triangular Banach algebra is permanently weakly module amenable (as an . As an example, we show that T0 is T0-module Arens regular if and only if the maximal group homomorphic image GS of S is finite.

In this paper we define the module topological center of the second dual \(\mathcal{A}^{**}\) of a Banach algebra \(\mathcal{A}\) which is a Banach \(\mathfrak{A}\)-module with compatible actions on another Banach algebra \(\mathfrak{A}\). We calculate the module topological center of ℓ 1(S)**, as an ℓ 1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. 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.

We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.

In this paper, we study the hereditary properties of module $(\phi,\varphi)$-amenability on Banach algebras. We also define the concept of module character contractibility for Banach algebras and obtain characterizations of module character contractible Banach algebras in terms of the existence of module $(\phi,\varphi)$-diagonals. We introduce module approximately character amenable Banach algebras. Finally, for every inverse semigroup $S$ with subsemigroup $E$ of idempotents, we find necessary and sufficient conditions for the $\ell^1(S)$ and its second dual to be module approximate character amenable (as a $\ell^1(E)$-module).

In this paper, we define the notion of sigma-approximate module amenability of Banach algebras and give some properties about this notion. Also for Banach A-bimodule A, and J, the closed ideal of A generated by elements of form ( α · a ) b - a ( b · α ) , ( a , b ∈ A , α ∈ A ) , we considered some corollaries about σ ^ -approximate amenability of A J as a Banach A-bimodule, where σ ^ : A J → A J by σ ^ ( a + J ) = σ ( a ) + J has a dense range.