Weak module amenability for the second dual of a Banach algebra

In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

Amenability of A⊕_T X as an extension of Banach algebra

Let 𝐴𝐴,𝑋𝑋,𝔘𝔘 be Banach algebras and 𝐴𝐴 be a Banach 𝔘𝔘-bimodule also 𝑋𝑋 be a Banach 𝐴𝐴−𝔘𝔘-module. In this paper we study the relation between module amenability, weak module amenability and module approximate amenability of Banach algebra 𝐴𝐴⊕𝑇𝑇𝑋𝑋 and that of Banach algebras 𝐴𝐴,𝑋𝑋. Where 𝑇𝑇: 𝐴𝐴×𝐴𝐴→𝑋𝑋 is a bounded bi-linear mapping with specificconditions.

Weak module amenability of triangular banach algebras II

Let A and B be Banach 𝔄-bimodule and Banach 𝔅-bimodule algebras, respectively. Also let M be a Banach A, B-module and Banach 𝔄, 𝔅-module with compatible actions. In the case of 𝔄 = 𝔅, the author along with Pourabbas [5] have studied the weak 𝔄-module amenability of triangular Banach algebra $\begin{array}{} \displaystyle \mathcal{T}=\left[\begin{array}{rr} A & M \\ & B \end{array} \right] \end{array}$ and showed that 𝓣 is weakly 𝔄-module amenable if and only if the corner Banach algebras A and B are weakly 𝔄-module amenable, where A, B and M are unital. In this paper we investigate a special structure of 𝔄 ⊕ 𝔅-bimodule derivation from 𝓣 into 𝓣∗ and show that 𝓣 is weakly 𝔄 ⊕ 𝔅-bimodule amenable if and only if the corner Banach algebras A and B are weakly 𝔄-module amenable and weakly 𝔅-module amenable, respectively, where A, B and M are essential and not necessary unital.

The first module (σ,τ)-cohomology group of triangular Banach algebras of order three

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson’s amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra [Formula: see text], where [Formula: see text] and [Formula: see text] are Banach algebras (with [Formula: see text]-module structure) and [Formula: see text] is a Banach [Formula: see text]-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949–958], and they showed that the weak module amenability of [Formula: see text] triangular Banach algebra [Formula: see text] (as an [Formula: see text]-bimodule) is equivalent with the weak module amenability of the corner algebras [Formula: see text] and [Formula: see text] (as Banach [Formula: see text]-bimodules). The main aim of this paper is to investigate the module [Formula: see text]-amenability and weak module [Formula: see text]-amenability of the triangular Banach algebra [Formula: see text] of order three, where [Formula: see text] and [Formula: see text] are [Formula: see text]-module morphisms on [Formula: see text]. Also, we give some results for semigroup algebras.

Permanent Weak Module Amenability of Semigroup Algebras

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.

n-Weak Module Amenability of Triangular Banach Algebras

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 -

Weak module amenability of triangular Banach algebras

Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra $\mathcal{T} = \left[ {_B^{AM} } \right]$ and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When $\mathfrak{A}$ is a Banach algebra and A and B are Banach $\mathfrak{A}$-module with compatible actions, and M is a commutative left Banach $\mathfrak{A}$-A-module and right Banach $\mathfrak{A}$-B-module, we show that A and B are weakly $\mathfrak{A}$-module amenable if and only if triangular Banach algebra T is weakly $\mathfrak{T}$-module amenable, where $\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} $.