Lau Product Research Articles

Norms on Lau product of normed algebras and their comparison

Norms on Lau product of normed algebras and their comparison

On amenable-like properties of some Banach algebras with applications to study their nilpotent ideals

In this current article, we show that a non-zero closed nilpotent ideal in a Johnson pseudo-contractible Banach algebra cannot be [Formula: see text]-predual. We prove also that a nilpotent ideal in a Johnson pseudo-contractible Banach algebras with the approximation property is forced to be zero. Among other things, we characterize the property [Formula: see text] of some semigroup algebras associated with inverse semigroups. More especially, for an inverse semigroup [Formula: see text] such that [Formula: see text] is uniformly locally finite, [Formula: see text] has the property [Formula: see text] if and only if each maximal subgroup of [Formula: see text] is amenable, where [Formula: see text] is the set of idempotents of [Formula: see text]. Furthermore, we study the property [Formula: see text] of some [Formula: see text]-Lau product structures.

Zero product determined of amalgamated duplication of Banach algebras

In this paper, we study some sufficient and necessary conditions for the amalgamated duplication Banach algebra [Formula: see text] to be zero product determined. Precisely, we show that in the case where [Formula: see text] has a bounded approximate identity, [Formula: see text] is zero product determined if and only if [Formula: see text] is zero product determined and [Formula: see text] has the property [Formula: see text]. Moreover, we give a better characterization when [Formula: see text] is unital. As an application, we also construct some examples on semidirect product of Banach algebras and the [Formula: see text]–Lau product of Banach algebras. Finally, we raise some open problems related to the zero product determined of classical module extension Banach algebras.

Spectrum, homomorphisms and multipliers of Lau product of Banach algebras

Given Banach algebras $A$, $B$ and a continuous homomorphism $\theta:B\longrightarrow A$ with $\Vert\theta\Vert \leq1$, we obtain characterization of spectrum, homomorphisms and multipliers of $A\times_{\theta}B$, which is a strongly splitting Banach algebra extension of $B$ by $A$. Also we characterize the semisimplicity of these algebras.

On approximately biprojective and approximately biflat Banach algebras

In this paper, we study the approximate biprojectivity and the approximate biflatness of a Banach algebra A and find some relations between theses concepts with ?-amenability and ? -contractibility, where ? is a character on A. Among other things, we show that ?-Lau product algebra L1(G) ?? A(G) is approximately biprojective if and only if G is finite, where L1(G) and A(G) are the group algebra and the Fourier algebra of a locally compact group G, respectively. We also characterize approximately biprojective and approximately biflat semigroup algebras associated with the inverse semigroups.

Diagonals of Lau product Banach algebras

In this paper, we study and compare the types of diagonals for Lau product Banach algebras [Formula: see text] and diagonals of two Banach algebras [Formula: see text] and [Formula: see text], where [Formula: see text] is a homomorphism from [Formula: see text] to [Formula: see text]. As a consequence, we obtain the (approximate) amenability and contractibility [Formula: see text] in terms of (approximate) amenability and contractibility of [Formula: see text] and [Formula: see text]. Moreover, we introduce the notion of pseudo-virtual diagonal for a Banach algebra and present some characterization results for Lau product Banach algebras.

On left φ-essential Connes-amenability of Banach algebras

In this paper, we introduce and study the notion of left [Formula: see text]-essential Connes amenable for dual Banach algebras. We investigate the hereditary properties of this new concept and we give some results for [Formula: see text]-Lau product and module extension. For unital dual Banach algebras, we show that left [Formula: see text]-essential Connes-amenability and left [Formula: see text]-Connes amenability are equivalent. Finally, with various examples, we examined this concept for upper triangular matrix algebras and [Formula: see text]-direct sum of Banach algebras.

Characterization of homological properties of θ-Lau product of Banach algebras

Let A and B be two Banach algebras and ?? ?(B): In this paper, we investigate biprojectivity and biflatness of ?-Lau product of Banach algebras A x? B. Indeed, we show that A x? B is biprojective if and only if A is contractible and B is biprojective. This generalizes some known results. Moreover, we characterize biflatness of ?-Lau product of Banach algebras under some conditions. As an application, we give an example of biflat Banach algebras A and X such that the generalized module extension Banach algebra X o A is not biflat. Finally, we characterize pseudo-contractibility of ?-Lau product of Banach algebras and give an affirmative answer to an open question.

Biprojectivity and Biflatness of Bi-amalgamated Banach Algebras

Let A, X be two Banach algebras and let X be an algebraic Banach A-module equipped with a bounded bilinear map $$\Theta :X\times X\rightarrow A$$ which is compatible with the A-module operations of X. Then the $$\ell ^1$$ -direct sum $$A\times X$$ endowed with the multiplication $$\begin{aligned} (a,x)(b,y)=(ab+\Theta (x,y),ay+xb+xy) \quad (a,b\in A, x, y\in X) \end{aligned}$$ is a Banach algebra, denoted by $$A\boxtimes _\Theta X$$ and will be called a bi-amalgamated Banach algebra. Many known Banach algebras such as (generalized) module extension Banach algebras, Lau product Banach algebras, generalized matrix Banach algebras have this general framework. The main aim of this paper is to investigate biprojectivity and biflatness of $$A\boxtimes _\Theta X$$ . Our results extend several results in the literature and provide simple direct proofs for some known results. In particular, we characterize the biprojectivity and biflatness of certain classes of the module extension Banach algebras and generalized matrix Banach algebras. Some unsolved questions are also included.

Some Notes on Semidirect Products of Banach Algebras

Let $${\mathcal {A}}$$ and $${\mathcal {U}}$$ be Banach algebras and $${\mathcal {A}} < imes {\mathcal {U}}$$ be the semidirect product of Banach algebras $${\mathcal {A}}$$ and $${\mathcal {U}}$$ where $${\mathcal {A}} < imes {\mathcal {U}}={\mathcal {A}}\oplus {\mathcal {U}}$$ as direct sum of Banach spaces and $${\mathcal {A}}$$ is closed subalgebra and $${\mathcal {U}}$$ is a closed ideal of $${\mathcal {A}} < imes {\mathcal {U}}$$ . The semidirect product $${\mathcal {A}} < imes {\mathcal {U}}$$ is a block of Banach algebras that covers the important classes of Banach algebras such as Lau product Banach algebras and triangular Banach algebras. In this paper we consider this question that under what conditions the semidirect product Banach algebra $${\mathcal {A}} < imes {\mathcal {U}}$$ is a Lau product Banach algebra or a triangular Banach algebra? We find these conditions and answer this question which form the main results of our paper.

$\Delta $-weak character amenability of certain Banach algebras

In this paper we investigate $\Delta $-weak character amenability of certain Banach algebras such as projective tensor product $A\widehat{\otimes }B$ and Lau product $A\times _{\theta }B$, where $A$ and $B$ are two arbitrary Banach algebras and $\theta \in \Delta (B)$, the character space of $B$. We also investigate $\Delta $-weak character amenability of abstract Segal algebras and module extension Banach algebras.

Some Homological Properties of Lau Product Algebras

For two Banach algebras A and B and for nonzero characters $$\theta $$ on B, the $$\theta $$ -Lau product is denoted by $$A\times _\theta B$$ . We study some homological properties of this Banach algebra such as Connes-amenability, projectivity, injectivity, flatness, approximate biflatness and $$\phi $$ -biprojectivity.

More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism

Abstract For two Banach algebras A and B, the T-Lau product A× T B, was recently introduced and studied for some bounded homomorphism T : B → A with ∥T∥ ≤ 1. Here, we give general nessesary and sufficent conditions for A× T B to be (approximately) cyclic amenable. In particular, we extend some recent results on (approximate) cyclic amenability of direct product A ⊕ B and T-Lau product A× T B and answer a question on cyclic amenability of A× T B.

English

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.

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.

Spectral properties of the Lau product $\mathcal{A}\times_{\theta}\mathcal{B}$ of Banach algebras

Let A and B be commutative Banach algebras. Then a multiplicative linear functional θ on B induces a multiplication on the Cartesian product space A×B given by (a,b)(c,d)=(ac+θ(d)a+θ(b)c,bd) for all (a,b),(c,d)∈A×B. We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn–Banach property, and the unique semisimple norm property under certain conditions on θ.

Generalized module extension Banach algebras: Derivations and weak amenability

Let A and X be Banach algebras and let X be an algebraic Banach A-module. Then the ℓ1-direct sum A × X equipped with the multiplicationis a Banach algebra, denoted by A ⋈ X, which will be called “a generalized module extension Banach algebra”. Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We characterize the structure of n-dual valued (n ∈ ℕ) derivations on A ⋈ X from which we investigate the n-weak amenability for the algebra A ⋈ X. We apply the results and the techniques of proofs for presenting some older results with simple direct proofs.

Derivations into various duals of Lau product of Banach algebras

Derivations into various duals of Lau product of Banach algebras

Approximate Cyclic Amenability of $T$-Lau Product of Banach Algebras

Associated with two Banach algebras $\mathcal{A}$ and $\mathcal{B}$ and a norm decreasing homomorphism $T: \mathcal{B} \to \mathcal{A}$, there is a certain Banach algebra product $\mathcal{M}_T := \mathcal{A} \times_T \mathcal{B}$, which is a splitting extension of $\mathcal{B}$ by $\mathcal{A}$. In this paper, we investigate approximate cyclic amenability of $\mathcal{M}_T$ which has been introduced and studied by Esslamzadeh and Shojaee in [5]. In particular, apart from the characterization of all cyclic derivations on the Banach algebra $\mathcal{M}_T$, we improve the results of [1, 2] for cyclic amenability of $\mathcal{M}_T$. These results paves the way for obtaining new results for (approximate) cyclic amenability of the Banach algebra $\mathcal{A} \times \mathcal{B}$ equipped with the coordinatewise product algebra and $\ell^1$-norm.

Some homological properties of $${T}$$ T -Lau product algebras

Let \({T}\) be a homomorphism from a Banach algebra \({B}\) to a Banach algebra \({A}\). The Cartesian product space \({A\times B}\) with \({T}\)-Lau multiplication and \({\ell^1}\)-norm becomes a new Banach algebra \({A\times _T B}\). We investigate the notions such as approximate amenability, pseudo amenability, \({\phi}\)-pseudo amenability, \({\phi}\)-biflatness and \({\phi}\)-biprojectivity for Banach algebra \({A\times_T B}\). We also present an example to show that approximate amenability of \({A}\) and \({B}\) is not stable for \({A\times _TB}\). Finally we characterize the double centralizer algebra of \({A\times _T B}\) and present an application of this characterization.