Module Amenability Research Articles

Various notions of module amenability on weighted semigroup algebras

Abstract Let S S be an inverse semigroup with the set of idempotents E E . In this article, we find necessary and sufficient conditions for the weighted semigroup algebra l 1 ( S , ω ) {l}^{1}\left(S,\omega ) to be module approximately amenable (contractible) and module character amenable (as l 1 ( E ) {l}^{1}\left(E) -module).

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.

Module bundles and module amenability

Let X be a compact Hausdorff space, and let {Ax : x ∈ X} and {Bx : x ∈ X} be collections of Banach algebras such that each Ax is a Bx-bimodule. Using the theory of bundles of Banach spaces as a tool, we investigate the module amenability of certain algebras of Ax-valued functions on X over algebras of Bx-valued functions on X.

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.

Ideal Module Amenability of Triangular Banach Algebras

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 Τ = [A M B ] (as a Τ = {[α α] : α ∈u}-module) and (2n - 1)-ideal module amenability of A and B (as an u-module), where u is a (not necessarily unital) Banach algebra such that A, B and M are commutative Banach u-bimodules. Finally, in the case that A = B = M = l 1 (S) and u = l 1 (E), for unital and commutative inverse semigroup S with idempotent set E, we show that T as an u-module is (2n - 1)- ideal module amenable while is not module amenable.

Biprojectivity and biflatness of amalgamated duplication of Banach algebras

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. We show that (super) amenability of [Formula: see text] implies (super) module amenability of [Formula: see text] and (super) amenability [Formula: see text]. We investigate biprojectivity and biflatness of [Formula: see text] in the some especial cases. We also give some results related to module biprojectivity and module biflatness of [Formula: see text], when [Formula: see text] is biprojective or biflat.

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.

Module amenability, module character biprojectivity and module character biflatness of lau product of two Banach algebras

Let A be a Banach algebra and T be an U-module homomorphism from U-bimodule B into U-bimodule A. We investigate module amenability (resp. module approximate amenability), module character amenability (resp. module character approximate amenability), module character biprojectivity and module character biflatness of A x Tu B for every two Banach U-bimodule A and B.

Symmetric module and connes amenability

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $ell^1(S)$ is a symmetric $ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigroup $E(S)$ of idempotents. In symmetric connes amenability, we have proved that $M(G)$ is symmetric connes amenable if and only if $G$ is amenable.

On homological properties of some module derivations on Banach algebras

In recent years, lots of papers have been published on module amenability. In this paper, our main aim is to study the homological properties of various module derivations and prove some results about module amenability. So this paper continous a line investigation in [3], [4] for Banach algebras.

N-Weak Module Amenability of Triangular Banach Algebras

Abstract 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.

Permanent Weak Module Amenability of Semigroup Algebras

Abstract 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.

Module amenability of restricted semigroup algebras under module actions

In this article, weshow that module amenability with the canonical action of restricted semigroup algebra l1r (S) and semigroup algebra l1(Sr) are equivalent, where Sr is the restricted semigroup of associated to the inverse semigroup S. We use this to give a characterization of module amenability of restricted semigroup algebra l1r (S) with the canonical action, where S is a Clifford semigroup.

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

In this paper, we generalize the notion of \(n\)-weak module amenability of a Banach algebra \(A\) which is a Banach module over another Banach algebra \(U\) with compatible actions to that of \((\sigma )-n\)-weak module amenability for \(n\in \mathbb {N}\) and \(\sigma \in Hom_{U}(A)\). We also investigate the relation between this new concept of amenability of \(A\) and the quotient Banach algebra \(A/J\) where \(J\) is the closed ideal of \(A\) generated by elements of the form \((a\cdot \alpha )b-a(\alpha \cdot b)\) for \(a,b\in A\) and \(\alpha \in U\). As a consequence, we show that the semigroup algebra \(l^{1}(S)\) is \((\sigma )\)-\((2n+1)\)-weakly module amenable as an \(l^{1}(E)\)-module for each \(n\in \mathbb {N}\) and \(\sigma \in Hom_{l^{1}(E)}(l^{1}(S))\), where \(S\) is an inverse semigroup with the set of idempotents \(E\).

σ -Approximately module amenable Banach algebras

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.

Super module amenability of inverse semigroup algebras

In this paper we compare the notions of super amenability and super module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. We find conditions for the two notions to be equivalent. In particular, we study arbitrary module actions of l1(ES) on l1(S) for an inverse semigroup S with the set of idempotents ES and show that under certain conditions, l1(S) is super module amenable if and only if S is finite. We also study the super module amenability of l1(S)∗∗ and module biprojectivity of l1(S), for arbitrary actions.

(Super) module amenability, module topological centre and semigroup algebras

For a Banach algebra \(\mathcal{A}\) which is also an \(\mathfrak{A}\)-bimodule, we study relations between module amenability of closed subalgebras of \(\mathcal{A}''\), which contains \(\mathcal{A}\), and module Arens regularity of \(\mathcal{A}\) and the role of the module topological centre in module amenability of \(\mathcal{A}''\). Then we apply these results to \(\mathcal{A}=l^{1}(S)\) and \(\mathfrak{A}=l^{1}(E)\) for an inverse semigroup S with subsemigroup E of idempotents. We also show that l 1(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group \(\frac{S}{\approx}\), is amenable. Moreover, we define super module amenability and show that l 1(S) is super module amenable if and only if \(\frac{S}{\approx}\) is finite.

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

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.

The structure of module amenable Banach algebras

In this paper, we study the module amenability of Banach al- gebras and characterize it in terms the concepts spliting and admissiblity of short exact sequences.

Weak Module Amenability for Semigroup Algebras

We study the weak module amenability of Banach algebras which are Banach module over another Banach algebra with compatible actions. As an example we show that the semigroup algebra of a commutative inverse semigroup is always weakly amenable as a module over the semigroup algebra of its subsemigroup of idempotents.