An amenable, radical Banach algebra

Different types of weak amenability for Banach algebras

In this paper, we introduce and investigate the concepts of cyclically weakly amenable and point amenable. Then, we compare these concepts with the concepts of weakly amenable and cyclically amenable and find the relation between them. For example, we prove that a Banach algebra is weakly amenable if and only if it is both cyclically amenable and cyclically weakly amenable. In the case where $A$ is commutative, the weak amenability and cyclically weak amenability of $A$ are equivalent. We also show that if $A$ is a Banach algebra with $Δ(A)\neq\emptyset$, then $A$ is cyclically weakly amenable…

Banach space properties forcing a reflexive amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If $A$ is a reflexive, amenable Banach algebra such that for each maximal left ideal $L$ of $A$ (i) the quotient $A / L$ has the approximation property and (ii) the canonical map from $A \check{\otimes} L^\perp$ to $(A / L) \wtensor L^\perp$ is open, then $A$ is finite-dimensional. As an application, we show that, if …

对称伪顺从 Banach 代数

Symmetrically pseudo-amenable Banach algebras

本文提出对称伪顺从性概念，刻画对称伪顺从 Banach 代数的性质、特例及其与其他顺从性的关系，并用于研究 Jordan 导子和 Lie 导子。

Topological radicals, IV. Frattini theory for Banach Lie algebras

The paper develops the theory of topological radicals of Banach Lie algebras and studies the structure of Banach Lie algebras with sufficiently many Lie subalgebras of finite codimensions -- the intersection of all these subalgebras is zero. It is shown that the intersections of certain families of Lie subalgebras (closed Lie subalgebras of finite codimension, closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) correspond to different preradicals, and that these preradicals generate the same radical, th…

On biamenability of Banach algebras

In this paper, we introduce the concept of biamenability of Banach algebras and we show that despite the apparent similarities between amenability and biamenability of Banach algebras, they lead to very different, and somewhat opposed, theories. In this regard, we show that commutative Banach algebras such as R and C tend to lack biamenability, while they may be amenable and highly noncommutative Banach algebras such as B(H) for an infinite dimensional Hilbert space H tend to be biamenable, while they are not amenable. Also, we show that although the unconditional unitization of an amenable Ba…

A new fixed point approach to hyperstability of radical-type functional equations in quasi-$(2,β)$-Banach spaces

The main focus of this paper is to define the notion of quasi-$(2,β)$-Banach space and show some properties in this new space, by help of it and under some natural assumptions, we prove that the fixed point theorem [16, Theorem 2.1] is still valid in the setting of quasi-$(2,β)$-Banach spaces, this is also an extension of the fixed point result of Brzdęk et al. [12, Theorem 1] in $2$-Banach spaces to quasi-$(2,β)$-Banach spaces. In the next part, we give a general solution of the radical-type functional equation (1.2). In addition, we study the hyperstability results for these functional equat…

A generalization of the weak amenability of some Banach algebra

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We show that by some new conditions, $A$ is weakly amenable whenever $A^{**}$ is weakly amenable. We will study this problem under generalization, that is, if $(n+2)-th$ dual of $A$, $A^{(n+2)}$, is $T-S-$weakly amenable, then $A^{(n)}$ is $T-S-$weakly amenable where $T$ and $S$ are continuous linear mappings from $A^{(n)}$ into $A^{(n)}$.

Homomorphisms, amenability and weak amenability of Banach algebras

In this paper we find some necessary and sufficient conditions for a Banach algebra to be amenable or weakly amenable, by applying the homomorphisms on Banach algebras.

Ideal amenability of Banach algebras on locally compact groups

In this paper we study the ideal amenability of Banach algebras. Let $\cal A$ be a Banach algebra and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $I$-weakly amenable if $H^{1}({\cal A},I^*)=\{0\}$. Further, $\cal A$ is ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenabil…

Approximate weak amenability of certain Banach algebras

It is shown that for a locally compact group $G$, if $L^{1}(G)^{**}$ is approximately weakly amenable, then $M(G)$ is approximately weakly amenable. Then, new notions of approximate weak amenability and approximate cyclic amenability for Banach algebras are introduced. Bounded $ω^{*}$-approximately weakly [cyclic] amenable $\ell^{1}$-Munn algebras are characterized.

Perturbations of Banach algebras and amenability

In this paper we prove that if (A,π) is an amenable Banach algebra and if ρis another Banach algebra multiplication on A such that the difference between ρand and πis less than 1/11, then (A, ρ) is also amenable.

Approximate characterizations of projectivity and injectivity for Banach modules

We characterize projective and injective Banach modules in approximate terms, generalizing thereby a characterization of contractible Banach algebras given by F. Ghahramani and R. J. Loy. As a corollary, we show that each uniformly approximately amenable Banach algebra is amenable. Some applications to homological dimensions of Banach modules and algebras are also given.

Approximate amenability of tensor products of Banach algebras

We show that the tensor product of approximately amenable algebras need not be approximately amenable, and investigate conditions under which $A$ and $B$ being approximately amenable implies, or is implied by, $A\hat{\otimes}B$ or $A^{\#}\hat{\otimes} B^{\#}$ being approximately amenable. Our methods also enable us to prove that if $A\hat{\otimes}B$ is amenable, then so are $A$ and $B$, a result proved by Barry Johnson in 1996 under an additional assumption.

Module approximate amenability of Banach algebras

In the present paper, the concepts of module (uniform) approximate amenability and contractibility of Banach algebras that are modules over another Banach algebra, are introduced. The general theory is developed and some hereditary properties are given. In analogy with the Banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same properties. It is also shown that module uniform approximate (contractibility) amenability and module (contractibility, respectively) amenability for commutative Banach modules are equivalent. Applying t…

On $Δ$-weak $ϕ$-amenability of Banach algebras

Let $A$ be a Banach algebra and $ϕ\in Δ(A)\cup\{0\}$. We say that $A$ is $Δ$-weak $ϕ$-amenable if there exists an $m\in A^{**}$ such that $m(ϕ)=0$ and $m(ψ.a)=ψ(a)$ for each $ψ\in Δ(A)$ and $a\in \ker(ϕ)$. It is shown that $A$ is $Δ$-weak $ϕ$-amenable if and only if $\ker(ϕ)$ has a bounded $Δ$-weak approximate identity. We examine this notion for some algebras over amenable locally compact groups. Also we prove that every $Δ$-weak $ϕ$-amenable Banach algebra has a bounded $Δ$-weak approximate identity.

$ϕ$-Johnson 可和 Banach 代数的 Jordan 导子与 Lie 导子

Jordan and Lie derivations of $ϕ$-Johnson amenable Banach algebras

本文研究 $ϕ$-Johnson 可和 Banach 代数在特定一维模作用下的导子结构，证明连续 Jordan 导子必为导子，连续 Lie 导子可分解为导子与中心值迹之和。

Relative amenability of Banach algebras

Let A be a Banach algebra and I be a closed ideal of A. We say that A is amenable relative to I, if A/I is an amenable Banach algebra. We study the relative amenability of Banach algebras and investigate the relative amenability of triangular Banach algebras and Banach algebras associated to locally compact groups. We generalize some of the previous known results by applying the concept of relative amenability of Banach algebras, especially, we present a generalization of Johnson's theorem in the concept of relative amenability.

Amenable crossed product Banach algebras associated with a class of $\mathrm{C}^\ast$-dynamical systems. II

We prove that the crossed product Banach algebra $\ell^1(G,A;α)$ that is associated with a ${\mathrm C}^\ast$-dynamical system $(A,G,α)$ is amenable if $G$ is a discrete amenable group and $A$ is a strongly amenable ${\mathrm C}^\ast$-algebra. This is a consequence of the combination of a more general result with Paterson's characterisation of strongly amenable unital $\mathrm{C}^\ast$-algebras in terms of invariant means for their unitary groups.

Approximate semi-amenability of Banach algebras

Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\rightarrow \mathcal{X}$ is approximately semi-inner derivation if there eixist nets $(ξ_α)_α$ and $(μ_α)_α$ in $\mathcal{X}$ such that, for each $a\in\mathfrak{A}$, $\mathcal{D}(a)=\lim_α(a.ξ_α-μ_α.a)$. $\mathfrak{A}$ is called approximately semi-amenable if for every Banach $\mathfrak{A}$-bimodule $\mathcal{X}$, every $\mathcal{D}\in\mathcal{Z}^{1}(\mathfrak{A},\mathcal{X}^{*})$ is approximtely semi-inner. There are some Banach algebras which are appro…

A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal

Let $G$ be a locally compact group, and let $WAP(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[SIN]$-group, but not compact, then the dual Banach algebra $WAP(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[SIN]$-group, $WAP(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal,virtual diagonal.

A generalization of cyclic amenability of Banach algebras

This paper continues the investigation of Esslamzadeh and the first author which was begun in [7]. It is shown that homomorphic image of an approximately cyclic amenable Banach algebra is again approximately cyclic amenable. Equivalence of approximate cyclic amenability of a Banach algebra $\mathcal{A}$ and approximate cyclic amenability of $M_{n}(\mathcal{A})$ is proved. It is shown that under certain conditions the approximate cyclic amenability of second dual $\mathcal{A}^{**}$ implies the approximate cyclic amenability of $\mathcal{A}$.

Derivation into duals of ideals of Banach algebras

We introduce two notions of amenability for a Banach algebra $\cal A$. Let $I$ be a closed two-sided ideal in $\cal A$, we say $\cal A$ is $I$-weakly amenable if the first cohomology group of $\cal A$ with coefficients in the dual space $I^*$ is zero; i.e., $H^1({\cal A},I^*)=\{0\}$, and, $\cal A$ is ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study the $I$-weak amenability of a Bana…

Module Pseudo-amenability of Banach algebras

The notions of module pseudo-amenable and module pseudo-contractible Banach algebras are introduced. For a Banach algebra with bounded approximate identity, module pseudo-amenability and module approximate amenability are the same properties. It is given a complete characterization of module pseudo-amenability for a Banach algebra. For every inverse semigroup $S$ with subsemigroup $E$ of idempotents, necessary and sufficient conditions are obtained for the $\ell^1(S)$ and its second dual to be $\ell^1(E)$-module pseudo-amenable.

Amenable semigroups of nonexpansive mappings on weakly compact convex sets

We show a few fixed point theorems for semigroups acting on weakly compact convex subsets of Banach spaces when $LUC(S), AP(S), WAP(S)$ or $WAP(S)\cap LUC(S)$ have a left invariant mean. In particular, we give a characterization of semitopological semigroups that have a left invariant mean on the space of weakly almost periodic functions in terms of a fixed point property for nonexpansive mappings. It answers, in the case of Banach spaces, Question 4 of [A.T.-M. Lau, Y. Zhang, J. Funct. Anal. 263 (2012), 2949--2977] in affirmative. We also extend the fixed point theorem of R. Hsu from left rev…

Derivations into n-th duals of ideals of Banach algebras

We introduce two notions of amenability for a Banach algebra $\cal A$. Let $n\in \Bbb N$ and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $n-I-$weakly amenable if the first cohomology group of $\cal A$ with coefficients in the n-th dual space $I^{(n)}$ is zero; i.e., $H^1({\cal A},I^{(n)})=\{0\}$. Further, $\cal A$ is n-ideally amenable if $\cal A$ is $n-I-$weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We find some relationships of $n-I-$ weak amenability and $m-J-$ weak amenability for some different m and n or for different closed ideals $I$ and $J$ of $\c…

$WAP$-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 enve…

On approximately left phi-biprojective Banach algebras

In this paper, for a Banach algebra A, we introduced the new notions of approximately left $ϕ$-biprojective and approximately left character biprojective, where $ϕ$ is a non-zero multiplicative linear functional on A. We show that for SIN group G, Segal algebra S(G) is approximately left $ϕ_1$- biprojective if and only if G is amenable, where $ϕ_1$ is the augmentation character on S(G). Also we showed that the measure algebra M(G) is approximately left character biprojective if and only if G is discrete and amenable. For a Clifford semigroup S, we show that `1(S) is approximately left characte…

Approximate semi-amenability of Banach algebras

In recent work of the authors the notion of a derivation being approximately semi-inner arose as a tool for investigating (approximate) amenability questions for Banach algebras. Here we investigate this property in its own right, together with the consequent one of approximately semi-amenability. Under certain hypotheses regarding approximate identities this new notion is the same as approximate amenability, but more generally it covers some important classes of algebras which are not approximately amenable, in particular Segal algebras on amenable SIN-groups.

Amenability for dual Banach algebras

We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We…

Strong pseudo-amenability of some Banach algebras

In this paper we introduce a new notion of strong pseudo-amenability for Banach algebras. We study strong pseudo-amenability of some Matrix algebras. Using this tool, we characterize strong pseudo-amenability of $\ell^{1}(S)$, provided that $S$ is a uniformly locally finite semigroup. As an application we show that for a Brandt semigroup $S=M^{0}(G,I)$, $\ell^{1}(S)$ is strong pseudo-amenable if and only if $G$ is amenable and $I$ is finite. We give some examples to show the differences of strong pseudo-amenability and other classical notions of amenability.

Gamma-bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π: G --> B(X) be a bounded representation. We show that if the set {π(t) : t \in G} is gamma-bounded then πextends to a bounded homomorphism w : C*(G) --> B(X) on the group C*-algebra of G. Moreover w is necessarily gamma-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G is equal to either the real numbers, the integers or the unit circle, and/or when X has prope…

Zero Lie product determined Banach algebras

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $φ\colon A\times A\to \mathbb{C}$ with the property that $φ(a,b)=0$ whenever $a$ and $b$ commute is of the form $φ(a,b)=τ(ab-ba)$ for some $τ\in A^*$. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras $L^1(G)$ with $G$ an amenable locally compact group are zero Lie product determined.

与一类C*-动力系统相关的顺从交叉积Banach代数

Amenable crossed product Banach algebras associated with a class of $\mathrm{C}^\ast$-dynamical systems

本文证明当离散群G顺从且C*-代数A为交换或有限维时，相应C*-动力系统的交叉积Banach代数ℓ^1(A,G,α)具有顺从性，并展望了后续研究方向。

Topological amenability and Köthe co-echelon algebras

We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). By using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii-Sheinberg's criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.

On homological notions of Banach algebras related to a character

In this paper, we countinue our work in \cite{11}. We show that $L^{1}(G,w)$ is $ϕ_{0}$-biprojective if and only if $G$ is compact, where $ϕ_{0}$ is the augmentation character. We introduce the notions of character Johnson amenability and character Johnson contractibility for Banach algebras. We show that $\ell^{1}(S)$ is pseudo-amenable if and only if $\ell^{1}(S)$ is character Johnson-amenable, provided that $S$ is a uniformly locally finite band semigroup. We give some conditions whether $ϕ$-biprojectivity ($ϕ$-biflatness) of $\ell^{1}(S)$ implies the finiteness (amenability) of $S$, respec…

Zero Lie product determined Banach algebras, II

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $φ\colon A\times A\to \mathbb{C}$ satisfying $φ(a,b)=0$ whenever $ab=ba$ is of the form $φ(a,b)=ω(ab-ba)$ for some $ω\in A^*$. We prove that $A$ has this property provided that any of the following three conditions holds: (i) $A$ is a weakly amenable Banach algebra with property $\mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from $A$ into $A^*$ is an inner derivation, (iii) $A$ is the algebra of all $n\times n$ matrices, where $n\ge 2$, …

Approximate weak amenability of Banach algebras

In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate $n$-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs. Our main result is that under some mild conditions on a given Banach algebra $\A$, if its second dual $\A^{**}$ is $(2n-1)$-weakly [respectively approximately/ approximately weakly/ approximately $n$-weakly] amenable, then so is $\A…

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 $I$, $\mathbb{M}_I(\mathbb{C})$ under this new notion is forced to have a finite index. Finally, we prov…

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct su…

Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule

Let $A$ be a dual Banach algebra with predual $A_\ast$ and consider the following assertions: (A) $A$ is Connes-amenable; (B) $A$ has a normal, virtual diagonal; (C) $A_\ast$ is an injective $A$-bimodule. For general $A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $B(G)$ in terms of $G$: For amenable, discrete $G…

On the positive commutator in the radical

In this paper we prove that a positive commutator between a positive compact operator $A$ and a positive operator $B$ is in the radical of the Banach algebra generated by $A$ and $B$. Furthermore, on every at least three-dimensional Banach lattice we construct finite rank operators $A$ and $B$ satisfying $AB\geq BA\geq 0$ such that the commutator $AB-BA$ is not contained in the radical of the Banach algebra generated by $A$ and $B$. These two results now completely answer to two open questions published in [4]. We also obtain relevant results in the case of the Volterra and the Donoghue operat…

On Connes amenability of upper triangular matrix algebras

In this paper, we study the notion of Connes amenability for a class of $I\times{I}$-upper triangular matrix algebra $UP(I,\mathcal{A})$, where $\mathcal{A}$ is a dual Banach algebra with a non-zero $wk^\ast$-continuous character and $I$ is a totally ordered set. For this purpose, we characterize the $ϕ$-Connes amenability of a dual Banach algebra $\mathcal{A}$ through the existence of a specified net in $\mathcal{A}\hat{\otimes}\mathcal{A}$, where $ϕ$ is a non-zero $wk^\ast$-continuous character. Using this, we show that $UP(I,\mathcal{A})$ is Connes amenable if and only if $I$ is singleton a…

Generalized notions of module character amenability

In this paper, we study the hereditary properties of module $(ϕ,φ)$-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 $(ϕ,φ)$-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 $…

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, $T(xy)=Ψ(T(x),T(y))$, where $T$ is a unital bounded operator acting from an amenable Banach algebra to a dual Banach algebra.

Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras

In this note, we study the Arens regularity of projective tensor product $A\hat{\otimes}B$ whenever $A$ and $B$ are Arens regular. We establish some new conditions for showing that the Banach algebras $A$ and $B$ are Arens regular if and only if $A\hat{\otimes}B$ is Arens regular. We also introduce some new concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property. Then if $A^{**}$ is weakly amenab…

Applications of amenable semigroups in operator theory

The paper deals with continuous homomorphisms $S \ni s \mapsto T_s \in L(E)$ of amenable semigroups $S$ into the algebra $L(E)$ of all bounded linear operators on a Banach space $E$. For a closed linear subspace $F$ of $E$, sufficient conditions are given under which there exists a projection $P \in L(E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space, sufficient conditions are given for the existence of an invertible operator $R \in L(E)$ such that all $R T_s R^{-1}$ are isometries. Also certain results on extending intertwining operators, renorming as well as on opera…

Approximate and pseudo-amenability of various classes of Banach algebras

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity. Among our other results, it is shown that the Fourier algebra of the free group on two generators is not approximately amenable. Further examples are obtained of ${\ell}^1$-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate amenability need not imply sequential approximate amenability. Results…

Amenability of ultraproducts of Banach algebras

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of $\mc A$ is Arens regular, and give some evidence that this is if and only if $\mc A$ is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebra. We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of $\mc A$ is amenable. We provide an abstract cha…

$Δ$-weak character amenability of certain Banach algebras

In this paper we introduce the notion of $Δ$-weak character amenable Banach algebras and investigate $Δ$-weak character amenability of certain Banach algebras such as projective tensor product $A\widehat{\otimes}B$, Lau product $A\times_TB$ for every Banach algebra $ A$ and $B$, where $T$ be a homomorphism from $B$ into $A$, abstract Segal algebras and module extension Banach algebras.

On Character Amenability of Banach Algebras

Associated to a nonzero homomorphism $φ$ of a Banach algebra $A$, we regard special functionals, say $m_φ$, on certain subspaces of $A^\ast$ which provide equivalent statements to the existence of a bounded right approximate identity in the corresponding maximal ideal in $A$. For instance, applying a fixed point theorem yields an equivalent statement to the existence of a $m_φ$ on $A^\ast$; and, in addition we expatiate the case that if a functional $m_φ$ is unique, then $m_φ$ belongs to the topological center of the bidual algebra $A^{\ast\ast}$. An example of a function algebra, surprisingly…

Application of topological radicals to calculation of joint spectral radii

It is shown that the joint spectral radius $ρ(M)$ of a precompact family $M$ of operators on a Banach space $X$ is equal to the maximum of two numbers: the joint spectral radius $ρ_{e}(M)$ of the image of $M$ in the Calkin algebra and the Berger-Wang radius $r(M)$ defined by the formula \[ r(M)=\underset{n\to\infty}{\limsup}(\sup\left\{ρ(a):a\in M^{n}\right\} ^{1/n}) . \] Some more general Banach-algebraic results of this kind are also proved. The proofs are based on the study of special radicals on the class of Banach algebras.

Invariant and hyperinvariant subspaces for amenable operators

There has been a long-standing conjecture in Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to every non-scalar amenable operator has a non-trivial hyperinvariant subspace and equivalent to every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two decompositions for amenable operators, which supporting the conjecture.

Derivations And Cohomological Groups Of Banach Algebras

Let $B$ be a Banach $A-bimodule$ and let $n\geq 0$. We investigate the relationships between some cohomological groups of $A$, that is, if the topological center of the left module action $π_\ell:A\times B\rightarrow B$ of $A^{(2n)}$ on $B^{(2n)}$ is $B^{(2n)}$ and $H^1(A^{(2n+2)},B^{(2n+2)})=0$, then we have $H^1(A,B^{(2n)})=0$, and we find the relationships between cohomological groups such as $H^1(A,B^{(n+2)})$ and $H^1(A,B^{(n)})$, spacial $H^1(A,B^*)$ and $H^1(A,B^{(2n+1)})$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude t…

Cohomological properties and Arens regularity of Banach algebras

In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$ where, for every $b\in B$, $L_{b}(a)=ba$ and $R_{b}(a)=a b…