Dual Banach Algebra Research Articles

Weak amenability for dual Banach algebras

Abstract A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

A new class of ideal Connes amenability

In this paper, we introduce the notion of σ−ideally Connes amenable for dual Banach algebras and give some hereditary properties for this new notion. We also investigate σ−ideally Connes amenability of ℓ1(G,ω). We show that if ω is a diagonally bounded weight function on discrete group G and σ is isometrically isomorphism of ℓ1(G,ω), then ℓ1(G,ω) is σ−ideally Connes amenable and so it is ideally Connes amenable.

Enveloping Dual Banach Algebras and Approximate Properties

Suppose that A is a Banach algebra and FA is its enveloping dual Banach algebra, we show that FA is approximately contractible (approximately amenable) if A has the same property. Also, we study the relation between the pseudoamenability of FA and the pseudoamenability of the second dual A∗∗ and we also characterize approximate biflatness and approximate biprojectivity of FA associated with approximate biflatness and approximate biprojectivity of the second dual A∗∗.

Almost multiplicative maps with respect to almost associative operations on Banach algebras

We prove an Ulam type stability result for a non-associative version of the multiplicativity equation, that is, T(xy)=Psi (T(x),T(y)), where T is a unital bounded operator acting from an amenable Banach algebra to a dual Banach algebra.

A COUNTEREXAMPLE TO A RESULT OF JABERI AND MAHMOODI

AbstractWe show that $\ell ^1(\mathbb {N}_\wedge )$ is $\varphi $ -amenable for each multiplicative linear functional $\varphi :\ell ^1(\mathbb {N}_\wedge )\rightarrow \mathbb {C}.$ This is a counterexample to the final corollary of Jaberi and Mahmoodi [‘On $\varphi $ -amenability of dual Banach algebras’, Bull. Aust. Math. Soc.105 (2022), 303–313] and shows that the final theorem in that paper is not valid.

Ideal Connes-Amenability of Certain Dual Banach Algebras

Ideal Connes-Amenability of Certain Dual 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.

Some homological properties of the enveloping dual Banach algebra

Some homological properties of the enveloping dual Banach algebra

On Connes amenable-like properties of dual Banach algebras based on w * w^{*} -character space

Abstract In this paper, the concept of left ϕ-approximate Connes amenability for a dual Banach algebra 𝒜 {\mathcal{A}} is studied, where ϕ is a weak * {{}^{*}} -continuous multiplicative linear functional on 𝒜 {\mathcal{A}} . Some characterizations for module extension dual Banach algebras and matrix algebras are given. Moreover, the notion of approximate left ϕ-Connes biprojectivity for dual Banach algebras is introduced and some concrete examples regarding this new notion are indicated.

Amalgamated duplication of Banach algebras from homological point of view

In a paper from 2002, Zhang [Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131–4151] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. Following his approach, we show that for amalgamated duplication of dual Banach algebras, we are able to choose appropriate conditions for weak Connes amenability of those Banach algebras which extend the Zhang’s result in a much more general setting. Moreover, apart from the characterization of minimal idempotents, we restrict our investigation to the study of injectivity, projectivity and flatness of amalgamated duplication of Banach algebras. Particularly, it is worth mentioning that our result paves the way for studying of homological properties and weak Connes amenability of those dual Banach algebras that have a semidirect product-like structure including Lau algebras.

On Amenability‐Like Properties of a Class of Matrix Algebras

In this study, we show that a matrix algebra is a dual Banach algebra, where is a dual Banach algebra and 1 ≤ p ≤ 2. We show that is Connes amenable if and only if I is finite, for every nonempty set I. Additionally, we prove that is always pseudo‐Connes amenable, for 1 ≤ p ≤ 2. Also, Connes amenability and approximate Connes biprojectivity are investigated for generalized upper triangular matrix algebras. Finally, we show that is approximately biflat if and only if is approximately biflat and I is a singleton.

ON -AMENABILITY OF DUAL BANACH ALGEBRAS

AbstractGeneralising the concept of injectivity, we study the notion of $\varphi $ -injectivity for dual Banach algebras. It provides a framework for studying $\varphi $ -amenability of enveloping dual Banach algebras.

Johnson pseudo-Connes amenability of dual Banach algebras

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group G,M(G) is Johnson pseudo-Connes amenable if and only if G is amenable. Also we show that for every non-empty set I,MI(C) under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.

Preduals and complementation of spaces of bounded linear operators

For Banach spaces [Formula: see text] and [Formula: see text], we establish a natural bijection between preduals of [Formula: see text] and preduals of [Formula: see text] that respect the right [Formula: see text]-module structure. If [Formula: see text] is reflexive, it follows that there is a unique predual making [Formula: see text] into a dual Banach algebra. This removes the condition that [Formula: see text] have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement [Formula: see text] in its bidual and [Formula: see text]-linear projections that complement [Formula: see text] in its bidual. It follows that [Formula: see text] is complemented in its bidual if and only if [Formula: see text] is (either as a module or as a Banach space). Our results are new even in the well-studied case of isometric preduals.

Weighted Semigroup Algebras as Dual Banach Algebras

A dual Banach algebra is a Banach algebra which is a dual space, and its multiplication is separately $$w^*$$-continuous. Among other things, the main objective of this paper is to study those conditions under which the weighted semigroup algebra $$l^1(S,\omega )$$ is a dual Banach algebra with predual $$c_0(S)$$. Moreover, our results illustrated how the properties of a weight function could lead to a different behavior from that of $$l^1(S)$$ as dual Banach algebras.

Approximate Connes-biprojectivity of dual Banach algebras

In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, approximate Connes amenability and [Formula: see text]-Connes amenability. We propose a criterion to show that certain dual triangular Banach algebras are not approximately Connes-biprojective. Next, we show that for a locally compact group [Formula: see text], the Banach algebra [Formula: see text] is approximately Connes-biprojective if and only if [Formula: see text] is amenable. Finally, for an infinite commutative compact group [Formula: see text], we show that the Banach algebra [Formula: see text] with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.

Dual Fréchet algebras: Connes amenability and (σwc)-virtual diagonals

A relation between Connes amenability of dual Banach algebras and the existence of ([Formula: see text])-virtual diagonals was investigated by Volker Runde. In this paper, we first introduce and study the notion of dual Fréchet algebras. We then study the weak[Formula: see text]-weakly continuous elements of Fréchet bimodules, and the notion of normality for dual spaces of Fréchet bimodules. We obtain some important results about these notions. Finally, we introduce and study amenability for dual Fréchet algebras and investigate the relation between this notion and the existence of ([Formula: see text])-virtual diagonals.

$sigma$-Connes Amenability and Pseudo-(Connes) Amenability of Beurling Algebras

In this paper, pseudo-amenability and pseudo-Connes amenability of weighted semigroup algebra $ell^1(S,omega)$ are studied. It is proved that pseudo-Connes amenability and pseudo-amenability of weighted group algebra $ell^1(G,omega)$ are the same. Examples are given to show that the class of $sigma$-Connes amenable dual Banach algebras is larger than that of Connes amenable dual Banach algebras.

WAP-biprojectivity of the enveloping dual Banach algebras

In this paper, we introduce a new notion of biprojectivity, called $WAP$-biprojectivity for $F(\mathcal{A})$, the enveloping dual Banach algebra associated to a Banach algebra $\mathcal{A}$. We find some relations between Connes biprojectivity, Connes amenability and this new notion. We show that, for a given dual Banach algebra $\mathcal{A}$, if $F(\mathcal{A})$ is Connes amenable, then $\mathcal{A}$ is Connes amenable. For an infinite commutative compact group $G$, we show that the convolution Banach algebra $F(L^2(G))$ is not $WAP$-biprojective. Finally, we provide some examples of the enveloping dual Banach algebras and we study their $WAP$-biprojectivity and Connes amenability.

Quasi-multipliers on Banach algebras related to locally compact semigroups

Let $${\mathcal {S}}$$ be a compactly cancellative foundation semigroup with identity and $$M_a({\mathcal {S}})$$ be its semigroup algebra. In this paper, we give some characterizations for $$ {{\mathfrak {Q}}}{{\mathfrak {M}}}(M_a({\mathcal {S}}))$$, the quasi-multipliers of $$M_a({\mathcal {S}})$$. It is shown that $$ {{\mathfrak {Q}}}{{\mathfrak {M}}}(M_a({\mathcal {S}}))$$ may be identified by $$M({\mathcal {S}})$$. We deal with the quasi-multipliers on the dual Banach algebra $$L_{0}^{\infty }({\mathcal {S}};M_a({\mathcal {S}}))$$ and prove that its quasi-multipliers is again $$M({\mathcal {S}})$$. We also discuss the bilinear mappings $${\mathfrak {m}} :M_a({\mathcal {S}})^{*} \times M_a({\mathcal {S}})^{*} \longrightarrow M_a({\mathcal {S}})^{*}$$ which commutes with translations and convolutions.