We introduce the notions of approximate Connes-amenability and approximate strong Connes-amenability for dual Banach algebras. Then we characterize these two types of algebras in terms of approximate normal virtual diagonals and approximate $σWC-$virtual diagonals. We investigate these properties for von Neumann algebras and measure algebras of locally compact groups. In particular we show that a von Neumann algebra is approximately Connes-amenable if and only if it has an approximate normal virtual diagonal. This is the ``approximate'' analog of the main result of Effros in [E. G. Effros, Amenability and virtual diagonals for von Neumann algebras, J. Funct. Anal. 78 (1988), 137-153].
We show that in general the concepts of approximate Connes-ameanbility and Connes-ameanbility are distinct, but for measure algebras these two concepts coincide. Moreover cases where approximate Connes-amenability of $\A^{**}$ implies approximate Connes-amenability or approximate amenability of $\A$ are also discussed.
Let $T$ be a Banach algebra homomorphism from a Banach algebra $\mathcal B$ to a Banach algebra $\mathcal A$ with $\|T\|\leq 1$. Recently it has been obtained some results about Arens regularity and also various notions of amenability of $\mathcal A\times_{T}\mathcal B$, in the case where $\mathcal A$ is commutative. In the present paper, most of these results have been generalized and proved for an arbitrary Banach algebra $\mathcal A$.
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.
We shall develop two notions of pointwise amenability, namely pointwise Connes amenability and pointwise $w^*$-approximate Connes amenability, for dual Banach algebras which take the $w^*$-topology into account. We shall study these concepts for the Banach sequence algebras $\ell^1(ω)$ and the weighted semigroup algebras $ \ell^{1}(\mathbb{N}_{\wedge},ω)$. For a weight $ω$ on a discrete semigroup $S$, we shall investigate pointwise amenability/Connes amenability of $\ell^1(S,ω)$ in terms of diagonals.
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.
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.
In this paper, we introduce $p$-amenability, bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals for a Banach algebra $\mathfrak{A}$ where $s$ is a non-zero element of algebraic center of $\mathfrak{A}$ that is denoted by $Z(\mathfrak{A})$. We show that if a Banach algebra $\mathfrak{A}$ is $p$-amenable then it has bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals and by this fact we prove that if the Banach algebra $\mathfrak{A}$ is unital then $p$-amenability and symmetric amenability are equivalent.
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.
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…
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.
For two Banach algebras $A$ and $B$, the $T$-Lau product $A\times_T B$, was recently introduced and studied for some bounded homomorphism $T:B\to A$ with $\|T\|\leq 1$. Here, we give general nessesary and sufficent conditions for $A\times_T B$ to be (approximately) cyclic amenable. In particular, we extend some recent results on (approximate) cyclic amenability of direct product $A\oplus B$ and $T$-Lau product $A\times_T B$ and answer a question on cyclic amenability of $A\times_T B$.
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.
We shall develop a notion of amenability for dual Banach algebras, namely weak Connes amenability, which will play the role that weak amenability does for usual Banach algebras
Given Banach algebras $ A $ and $ B $ with $ θ\inΔ(B) $. We shall study the Johnson pseudo-contractibility and pseudo-amenability of $ θ$-Lau product $ A\times_θ B $. We show that if $ A\times_θ B $ is Johnson pseudo-contractible, then $ A $ is Johnson pseudo-contractible and has a bounded approximate identity and $ B $ is Johnson pseudo-contractible. In some particular cases complete characterization of Johnson pseudo-contractibility of $ A\times_θ B $ are given. Also, we show that pseudo-amenability of $ A\times_θ B $ implies approximate amenability of $ A $ and pseudo-amenability of $ B $.
In this paper, we introduce the new notion of strong pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion to the various notions of Connes amenability. Also we show that for every non-empty set $I$, $M_I(\mathbb{C})$ is strong pseudo-Connes amenable if and only if $I$ is finite. We provide some examples of certain dual Banach algebras and we study its strong pseudo-Connes amenability. In the last section, we investigate the property ultra central approximate identity for a Banach algebra $\mathcal{A}$ and its second dual $\mathcal{A}^{**}$. Also we …
Let $X$ be a compact Hausdorff space and $A$ a Banach algebra. We investigate amenability properties of the algebra $C(X,A)$ of all $A$-valued continuous functions. We show that $C(X,A)$ has a bounded approximate diagonal if and only if $A$ has a bounded approximate diagonal; if $A$ has a compactly central approximate diagonal (unbounded) then $C(X,A)$ has a compactly approximate diagonal. Weak amenability of $C(X,A)$ for commutative $A$ is also considered.
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…
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…
For a Banach algebra $ \mathcal{A} $, we introduce various approximate virtual diagonals such as approximate WAP-virtual diagonal and approximate virtual diagonal. For the enveloping dual Banach algebra $ F(\mathcal{A}) $ of $ \mathcal{A} $, we show that $ F(\mathcal{A}) $ is approximately Connes-amenable if and only if $ \mathcal{A} $ has an approximate WAP-virtual diagonal. Further, for a discrete group $ G $, we show that if the group algebra $ \ell^1(G) $ has an approximate WAP-virtual diagonal, then it has an approximate virtual diagonal.
Let $A$ and $X$ be Banach algebras and let $X$ be an algebraic Banach $A-$module. Then the $\ell^1-$direct sum $A\times X$ equipped with the multiplication $$(a,x)(b,y)=(ab, ay+xb+xy)\quad (a,b\in A, x,y\in X)$$ is a Banach algebra, denoted by $A\bowtie X$, which will be called "\textit{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 ($n\in\mathbb N)$) derivations on $A\bowtie X$ from which we investigate the $n-$we…
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 $…
In this paper, we introduce a new notion of amenability, $σ-$Connes ideal, say, for a large class of dual Banach algebras. We extend the concept of ideal Connes amenability and study their properties. Let $σ$ be a $weak^{*}$-continuous endomorphism on a dual Banach algebra $\mathcal{A}$ with dense range. Then the concept of ideal Connes-amenability and $σ-$ ideally Connes amenability are the same. We gave some general results and hereditary properties with some examples for this new notion of amenability.
Finie dimensional invariant subspace property and amenability for a class of Banach algebras
受Ky Fan在1965年的一个结果的启发，我们利用有限维不变子空间的性质，建立了左顺从F-代数（包括局部紧群的群代数和Fourier代数，量子群代数，或者更一般地说，Hopf von Neumann代数的前对偶代数）的一个刻画.这是通过首先揭示一个不动点性质的半群的范数为1的正线性泛函的代数。我们的结果回答了第一作者于1993年在东京提出的一个公开问题（见[25，问题5]）。我们还证明了F-代数中理想的左顺从性可以决定该代数的左顺从性。
Amenability modulo an ideal of a Banach algebra have been defined and studied. In this paper we introduce the concept of amenability modulo an ideal of a Frechet algebra and investigate some known results about amenability modulo an ideal of a Frechet algebra. Also we show that a Frechet algebra A is amenable modulo an ideal if and only if A is isomorphic to a reduced limit of amenable modulo an ideal of Banach algebras.
Let T be a homomorphism from a Banach algebra B to a Banach algebra A.The Cartesian product space A * B with T-Lau multiplication and l^1-norm becomes a new Banach algebra A *_T B. We investigate the notions such as approximate amenability, pseudo amenability, phi-pseudo amenability,phi-biflatness and phi-biprojectivity for Banach algebra A *_T B. We also present an example to show that approximate amenability of A and B is not stable for A *_T B. Finally we characterize the double centralizer algebra of A *_T B and present an application of this characterization.
Extending M.\ Daws' definition of ultra-amenable Banach algebras, we introduce the notion of operator ultra-amenability for completely contractive Banach algebras. For a locally compact group $G$, we show that the operator ultra-amenability of $A(G)$ imposes severe restrictions on $G$. In particular, it forces $G$ to be a discrete, amenable group with no infinite abelian subgroups. For various classes of such groups, this means that $G$ is finite.
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.
Associated with two Banach algebras $\mathcal A$ and $\mathcal B$ and a norm decreasing homomorphism $T:{\mathcal B}\rightarrow{\mathcal A}$, there is a certain Banach algebra product ${\mathcal A}\times_T {\mathcal B}$, which is a splitting extension of $\mathcal B$ by $\mathcal A$. We investigate some notions of amenability such as approximate weak amenability, approximate cyclic amenability, ideal amenability, approximate Connes-amenability, Pseudo-Connes amenability, $w^*$-approximate Connes-amenability, essential amenability, essential $ϕ$-amenability and essential character amenability. …
