Character Amenability Research Articles

Character amenability of vector-valued algebras

Let {Ax : x ∈ X} be a collection of complex Banach algebras indexed by the compact Hausdorff space X. We investigate the character amenability of certain algebras of Ax-valued functions in relation to the character amenability of the Ax.

On character amenability of the Fourier algebra

On character amenability of the Fourier algebra

Bounded version of approximate module character amenability of Banach algebras

The bounded version of approximate module character amenability of Banach algebras is introduced and studied. This new concept is characterized by several different concepts such as bounded approximate module character means. Moreover, this new concept is investigated for second dual, unitization, tensor product and lp-direct sums of Banach algebras.

Essential $$\varphi $$-amenability and essential character amenability of Fréchet algebras

Essential $$\varphi $$-amenability and essential character amenability of Fréchet algebras

Character amenability and character pseudo-amenability of certain Banach algebras

Character amenability and character pseudo-amenability of certain Banach algebras

Turán’s inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by L1-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated L1-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties hold (which is very different to L1-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Turán’s inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Turán type inequalities and transformations into more convenient orthogonal polynomial systems.

Essential character amenability of semigroup algebras

Let S be a foundation topological semigroup and $$M_a(S)$$ the space of all measures $$\mu \in M(S)$$ for which the maps $$x\longmapsto |\mu |*\delta _{x}$$ and $$x\longmapsto \delta _{x}*|\mu |$$ from S into M(S) are weakly continuous. In the present paper, we introduce and study the concept of $$\phi$$ -amenability for S and investigate the relations between $$\phi$$ -amenability of S and essential $$\widehat{\phi }$$ -amenability of $$M_a(S)$$ , where $$\phi$$ is a character on S and $$\widehat{\phi }$$ is the extension of $$\phi$$ to $$M_a(S)$$ .

Character amenability of real Banach algebras

‎Let $ (A,| cdot |) $ be a real Banach algebra‎. ‎In this paper we first introduce left and right $varphi$-amenability of $A$ and discuss the relation between left (right‎, ‎respectively) $varphi$-menability and $overline{varphi}$-amenability of $A$ for $varphiintriangle(A)cup{0}$ where $overline{varphi}intriangle(A)$ is the conjugate of $varphi$‎. ‎Next we show that $A$ is left (right‎, ‎respectively) $varphi$-amenable if and only if‎ ‎$A_{mathbb{C}}$ is left (right‎, ‎respectively) $varphi_{mathbb{C}}$-amenable‎, ‎where $A_{mathbb{C}}$ is a suitable complexification of $ A $ and $varphi_{mathbb{C}}intriangle(A_{mathbb{C}})$ is the induced character by‎ ‎$varphi$ on $A_{mathbb{C}}$‎. ‎In continue‎, ‎we give a hereditary property for 0-amenability of $A$‎. ‎We also study relations between the injectivity of Banach left $A$-modules and right $varphi$-amenability of $A$‎. ‎Finally‎, ‎we characterize the left character amenability of certain real Banach algebras‎.

Characterization of ^ϕ-amenability and ^ϕ-module amenability of semigroup algebras

For every inverse semigroup $S$ with subsemigroup $E$ of idempotents, necessary and sufficient conditions are obtained for the semigroup algebra $l ^{1}(S)$ to be $hat{phi}$-amenable and $hat{phi}$-module amenable. Also, we characterize the character amenability of semigroup algebra $l^1(S)$.

Bounded Approximate Character Amenability of Banach Algebras

The bounded approximate version of $varphi$-amenability and character amenability are introduced and studied. These new notions are characterized in several different ways, and some hereditary properties of them are established. The general theory for these concepts is also developed. Moreover, some examples are given to show that these notions are different from the others. Finally, bounded approximate character amenability of some Banach algebras related to locally compact groups are investigated.

Approximate essential character amenability of Banach algebras

Let A be a Banach algebra and ' be a character on A. The notions of approximate essential '-amenability and approximate essential character '-amenability of A are introduced and some properties of such algebras are investigated. Then by means of some examples the distinctions between this new notions and essentially -amenability for A are shown.

On character amenability of semigroup algebras

The main purpose of this paper is to investigate the character amenability of semigroup algebras. In this regard, the new concept character amenability modulo an ideal of Banach algebras are introduced. For a large class of semigroups such as E-inversive E-semigroup and eventually inverse semigroups, it is shown that the semigroup S is amenable if and only if the semigroup algebra $$l^1(S)$$ is character amenable modulo an ideal. Some characterizations of character amenability modulo an ideal of Banach algebras are studied and interesting examples are presented.

$\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.

Invariant $\varphi$-means for abstract Segal algebras related to locally compact groups

In this paper, for a locally compact group ${\mathcal G}$ we characterize character amenability and character contractibility of abstract Segal algebras with respect to the group algebra $L^1({\mathcal G})$ and the generalized Fourier algebra $A_p({\mathcal G})$. As a main result we prove that ${\mathcal G}$ is discrete and amenable if and only if some class of abstract Segal algebras in $L^1({\mathcal G})$ is character amenable. We also prove a similar result for abstract Segal algebras in $A_p({\mathcal G})$ and $C_0({\mathcal G})$. Finally, under some conditions we investigate when a commutative, semisimple, Tauberian Banach algebra is an ideal in its second dual space.

Pseudo-amenability and pseudo-contractibility of restricted semigroup algebra

Abstract In this article the pseudo-amenability and pseudo-contractibility of restricted semigroup algebra l r 1 ( S ) $l_r^1(S)$ and semigroup algebra, l 1(Sr) on restricted semigroup, Sr are investigated for different classes of inverse semi-groups such as Brandt semigroup, and Clifford semigroup. We particularly show the equivalence between pseudo-amenability and character amenability of restricted semigroup algebra on a Clifford semigroup and semigroup algebra on a restricted semigroup. Moreover, we show that when S = M 0(G, I)is a Brandt semigroup, pseudo-amenability of l 1(Sr) is equivalent to its pseudo-contractibility.

ϕ-amenability and character amenability of Fréchet algebras

Abstract Right φ-amenability and right character amenability have been introduced for Banach algebras. Here, these concepts will be generalized for Fréchet algebras. Then some of the previous available results about right φ-amenability and right character amenability for the case of Banach algebras will be verified for Fréchet algebras. Related results about Segal Fréchet algebras are provided.

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.

Character Amenability of the Intersection of Lipschitz Algebras

AbstractLet (X, d) be ametric space and letJ⊆ [0,∞) be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras and define a special Banach subalgebra of ∩y∊JLipyX, denoted by ILipJX. Mainly, we investigate theC-character amenability of ILipJX, in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap and obtain a necessary and suõcient condition forC-character amenability of ILipJX, specially Lipschitz algebras, under an additional assumption.

Generalized notions of module character amenability

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).

Character Connes-amenability of dual Banach algebras

The concept of left character Connes-amenability for a dual Banach algebra $${\mathcal {A}}$$ is introduced. We obtain a cohomological characterization of left character Connes-amenability as well as the relation between left $$\varphi $$ -Connes-amenability and existence of left $$\varphi $$ -normal virtual diagonals for a $$\omega ^{*}$$ -continuous character $$\varphi $$ . We prove that left character amenability of $${\mathcal {A}}$$ is equivalent to left character Connes-amenability of $${\mathcal {A}}^{**}$$ when $${\mathcal {A}}$$ is Arens regular. Moreover for a locally compact group G, we show that M(G) is left character Connes-amenable. In addition by means of some examples we show that for the new notion, the corresponding class of dual Banach algebras is larger than Connes-amenable dual Banach algebras.