Perturbations of Banach algebras and amenability
Miad Makareh Shireh
本文证明了：如果（A，π）是顺从的巴拿赫代数，并且如果ρ是A上的另一个巴拿赫代数的乘法，使得ρ与π之差小于1/11，则（A，ρ）也是顺从的.
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.
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Homomorphisms, amenability and weak amenability of Banach algebras
本文利用巴拿赫代数上的同态，给出了Banach代数服从或弱服从的充要条件。
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.
Arens regularity of module actions and weak amenability of Banach algebras
对于Banach左、右模作用，我们建立了模作用的拓扑中心之间的关系，得到了巴拿赫代数弱顺从性的一些结果。
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.
Generalization of Weak Amenability of Banach Algebras
弱顺从性的新概念，$(φ,ψ)$ -弱顺从性最近推出。本文考虑了这一新概念，并研究了群代数$M(G)$，$L^1(G)$和$S^1(G)$的$(φ,ψ)$ -弱顺从性.
A generalization of cyclic amenability of Banach algebras
本文继续了文[7]中开始的对拉姆扎德和第一作者的研究。证明了近似循环顺从巴拿赫代数的同态像也是近似循环顺从的.证明了巴拿赫代数$\mathcal{A}$的近似循环顺从性与$M_{n}(\mathcal{A})$的近似循环顺从性的等价性.结果表明，在一定条件下，第二个对偶$\mathcal{A}^{**}$的近似循环顺从性隐含着$\mathcal{A}$的近似循环顺从性。
Equivalence of Pseudo- and Approximate Amenability in Banach Algebras
本文证明了具有有界近似恒等元的Banach代数中伪可近似性蕴含近似可近似性，从而等价，并填补了先前文献中的证明空白。
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.
Character Inner Amenability of Certain Banach Algebras
研究了一类由投射张量积$A\hat{\otimes}B$、Lau积$A\times_θB$和模扩张$A\oplus X$组成的巴拿赫代数的特征标内相容性.文中还列举了一些有启发性的例子。
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…
Solved and unsolved problems in generalized notions of amenability for Banach algebras
本文综述了巴拿赫代数的（有界的、序列的）近似顺从性/可缩性和伪顺从性/可缩性的研究进展。我们将讨论有关这些概念的核心问题，并解决任何解决方案对该领域发展的重要性。还包括一些新的结果。
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 $A\times_T B$.
Constructions preserving $n$-weak amenability of Banach algebras
满射有界同态一般不能保持$n$ -弱顺从性.然而，我们表明，它保持的性质，如果涉及的同态享有一个右逆。我们研究这一事实的某些同态几个巴拿赫代数。
Module approximate amenability of Banach algebras
本文引入了巴拿赫代数上模的巴拿赫代数的模（一致）逼近顺从性和可压缩性的概念。发展了一般理论，并给出了一些遗传性质。通过与Banach代数近似顺从性的类比，证明了模近似顺从性与模压缩性是相同的性质。还证明了交换Banach模的模一致近似（可收缩性）顺从性和模（分别可收缩性）顺从性是等价的。将这些结果应用于作为$\ell^1(E)$ -模的$\ell^1(S)$，对于具有幂等元集$E$的逆半群$S$，证明了$\ell^1(S)$是模近似顺从（可收缩）的当且仅当它是模一致近似顺从的当且仅当$S$是顺从的.此外，$\ell^1(S)^{**}$是模（一致）近似顺从的当且仅当$S$的极大群同态像是有限的。
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
Ideal amenability of Banach algebras on locally compact groups
本文研究了巴拿赫代数的理想顺从性。设$\cal A$是巴拿赫代数，$I$是$\cal A$中的闭双边理想，$\cal A$是$I$ -弱顺从的，如果$H^{1}({\cal A},I^*)=\{0\}$ .此外，如果$\cal A$是$I$，则$\cal A$是理想服从的-对于$\cal A$中的每个闭合双边理想$I$弱服从。我们知道顺从的巴拿赫代数的连续同态像也是顺从的。我们证明了理想顺从性的同态性质的适当的直接和是真的类似于弱顺从性，我们应用这个结果的理想顺从性的局部紧群上的巴拿赫代数。
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 $.
Connes amenability of the second dual of Arens regular Banach algebras
本文研究了Arens正则巴拿赫代数的第二对偶的Connes顺从性。当然，我们对Volker Runde提出的问题提供了部分答案。当Arens正则模扩张巴拿赫代数是自反模时，给出了其第二对偶是Connes顺从的充要条件.
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 …
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。
Lifting derivations and $n-$weak amenability of the second dual of a Banach algebra
证明了对$n\geq 2$，巴拿赫代数$\A$的第二对偶$\A^{**}$的$n-$弱顺从性蕴涵$\A$的弱顺从性.我们还提供了一个肯定的答案的情况下，$n=1,$锐化一些旧的结果。我们的证明方法也提供了一个统一的方法来给一些已知的结果的情况下，$n=1$短的证明。
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…
n-Weak Module Amenability of Triangular Banach Algebras
设$\mathcal A$，$\mathcal B$是具有相容作用的Banach $\mathfrak A$ -模，$\mathcal M$是左Banach $\mathcal A$ - $\mathfrak A$ -模和右Banach $\mathcal B$ - $\mathfrak A$ -模.本文研究了三角Banach代数$\mathcal T=[ {cc} \mathcal A & \mathcal M & \mathcal B ]$（作为$\mathfrak T:=\Big{[ {cc} α& & α ] | α\in\mathfrak A\Big}$ -模）的模顺从性、$n$ -弱模顺从性和模Arens正则性.我们利用这些结果证明了：对于一个逆半群$S$，其子半群$E$是幂等元的，三角Banach代数$\mathcal T_0=[ {cc} \ell^1(S)& \ell^1(S) & \ell^1(S) ]$是永久弱模顺从的（作为$\mathfrak T_0=[ {cc} \ell^1(E)& & \ell^1(E) ]$ -模）.作为例子，我们证明了$\mathcal T_0$是$\mathfrak T_0$ -模Arens正则的当且仅当$S$的极大群同态象$G_S$是有限的.
Derivation into duals of ideals of Banach algebras
本文引入了巴拿赫代数$\cal A$的两个顺从性概念。设$I$是$\cal A$中的闭双边理想，我们说$\cal A$是$I$ -弱顺从的，如果$\cal A$的第一上同调群的系数在对偶空间$I^*$中为零;即，$H^1({\cal A},I^*)=\{0\}$，并且，如果$\cal A$是$I$，则$\cal A$是理想服从的-对于$\cal A$中的每个闭合双边理想$I$弱服从。我们将这些概念与巴拿赫代数的弱顺从性联系起来。我们还证明了理想顺从性不同于顺从性和弱顺从性。研究了巴拿赫代数$\cal A$对某些特殊的闭双边\break理想$I$的$I$ -弱顺从性.
Approximate Connes-amenability of dual Banach algebras
引入了对偶巴拿赫代数的近似Connes-相容性和近似强Connes-相容性的概念。然后用近似正规虚对角线和近似$σWC-$虚对角线刻画了这两类代数。我们研究了局部紧群的vonNeumann代数和测度代数的这些性质。特别地，我们证明了冯诺依曼代数是近似Connes-顺从的当且仅当它有一个近似正常的虚拟对角线。这是Effros在[E. G.冯·诺依曼代数的顺从性和虚拟对角线，J. Funct。Anal. 78（1988），137-153]。我们证明了在一般情况下，近似Connes-可名性和Connes-可名性的概念是不同的，但对于测度代数，这两个概念是一致的。此外，还讨论了$\A^{**}$的近似Connes可变性蕴涵$\A$的近似Connes可变性或近似可变性的情形.
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.
A generalization of the weak amenability of some Banach algebra
设$A$是一个巴拿赫代数，$A^{**}$是它的第二对偶，通过一些新的条件证明了当$A^{**}$是弱顺从的时，$A$是弱顺从的.我们将在推广下研究这个问题，即如果$A$的$(n+2)-th$对偶$A^{(n+2)}$是$T-S-$弱顺从的，则$A^{(n)}$是$T-S-$弱顺从的，其中$T$和$S$是从$A^{(n)}$到$A^{(n)}$的连续线性映射.
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…
Amenability of algebras of approximable operators
给出了巴拿赫空间上可逼近算子的巴拿赫代数顺从的一个充要条件。我们进一步调查这个代数和因式分解的运营商之间的关系，加强已知的结果和发展新的技术，以确定是否或不一个给定的巴拿赫空间进行一个顺从的代数的逼近运营商。使用这些技术，我们能够显示，除其他事项外，非顺从的代数逼近算子Tsirelson的空间。
Amenability of Banach algebras of compact operators
本文研究了巴拿赫空间X上紧算子的巴拿赫代数K（X）服从的条件。我们给出了X的一个对称化逼近性质，并证明了它是这样一个条件。这一性质被包括所有经典空间在内的广泛的巴拿赫空间所满足。然后，我们调查新的巴拿赫空间的建设，从旧的保持财产进行顺从代数的紧算子。粗略地说，对偶空间、前对偶空间和某些张量积确实继承了这个性质，而直和则没有。对于直和，这个问题与线性算子的因式分解密切相关。在最后一节中，我们讨论了一些开放的问题，特别是，什么性质的X是由K（X）的顺从性暗示的逆问题。
Approximate weak amenability of Banach algebras
本文讨论了四种广义的顺从性概念：近似顺从性、近似弱顺从性、近似循环顺从性和近似$n$ -弱顺从性。前两个是由Ghahramani和Loy在[9]中介绍和研究的。我们介绍了第三和第四个，我们通过一些例子，他们的区别与他们的经典类似物。我们的主要结果是：在一定的条件下，若巴拿赫代数$\A$的第二对偶$\A^{**}$是$(2n-1)$ -弱[分别为近似/近似弱/近似$n$ -弱]顺从的，则$\A$也是.同样，如果$\A$近似为$(n+2)$ -弱顺从，则它近似为$n$ -弱顺从。此外，我们证明了近似迹扩张性质和近似弱[分别循环]顺从性之间的关系。这回答了[9]中关于近似弱顺从性和循环顺从性的问题9.1。
Notes on amenability
证明了对于具有有界逼近恒等式的巴拿赫代数$A$，A与A的射影张量积的顺从性、A与A^{op}的射影张量积的顺从性和A的顺从性是等价的.若A是交换巴拿赫代数B中的闭理想，则A与B的投射张量积的弱顺从性蕴涵了A的弱顺从性.
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.
Module super-amenability for semigroup algebras
设$S$是具有幂等元集$E$的逆半群.本文定义了一个巴拿赫代数的模超顺从性，并证明了当$E$是向上定向的且从左到右作用于$S$时，半群代数$ \ell ^{1}(S)$是$\ell^{1}(E)$ -模超顺从的当且仅当$S$的适当群同态像是有限的.
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.
Generalized notions of character amenability
本文引入了特征标可缩性、近似特征标可变性（可缩性）和一致近似特征标可变性（可缩性）的概念。研究了巴拿赫代数的广义特征标顺从性概念之间的关系。我们证明了近似特征标顺从性和近似特征标收缩性是相同的性质，一致近似特征标顺从性和特征标顺从性也是相同的性质，一致近似特征标收缩性和特征标收缩性也是相同的性质。对于交换巴拿赫代数，我们证明了特征标可压缩性与可压缩性是相同的性质。此外，这些概念的一般理论的发展。
Approximate and pseudo-amenability of various classes of Banach algebras
我们继续调查的概念，近似顺从的工作中介绍的第二和第三作者。证明了每一个有界近似可缩巴拿赫代数都有一个有界近似恒等式.在我们的其他结果中，它表明，两个生成元上的自由群的傅立叶代数是不近似顺从的。进一步的例子得到的${\ell}^1$ -半群代数，近似顺从，但不顺从，使用这些，我们表明，有界近似顺从性不需要暗示顺序近似顺从性。给出了$L^1(G)$的Segal子代数，其中$G$是局部紧群，以及离散群$Γ$上的$p$ -伪函数的代数$PF_p(Γ)$（其中约化$C^*$ -代数是其特例）.
Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras
本文研究了当$A$和$B$是Arens正则的时，射影张量积$A\hat{\otimes}B$的Arens正则性.给出了巴拿赫代数$A$和$B$是Arens正则的充要条件是$A\hat{\otimes}B$是Arens正则的.引入了左弱$^*$ -弱收敛性质[ $Lw^*wc-$性质]和右弱$^*$ -弱收敛性质[ $Rw^*wc-$性质]的概念，并对Banach代数$A$，设$A^*$和$A^{**}$分别具有$Rw^*wc-$性质和$Lw^*wc-$性质.然后，如果$A^{**}$是弱顺从的，则可以得出$A$是弱顺从的。我们还提供了一些结果，这些性质之间的关系与一些特殊的导子$D:A\rightarrow A^*$。得到了巴拿赫代数的Arens正则性的一些结论。
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.
On $Δ$-weak $ϕ$-amenability of Banach algebras
设$A$是一个Banach代数，$ϕ\in Δ(A)\cup\{0\}$ .我们说$A$是$Δ$ -弱$ϕ$ -顺从的，如果存在一个$m\in A^{**}$，使得每个$ψ\in Δ(A)$和$a\in \ker(ϕ)$的$m(ϕ)=0$和$m(ψ.a)=ψ(a)$。证明了$A$是$Δ$ -弱$ϕ$ -顺从的当且仅当$\ker(ϕ)$有界$Δ$ -弱近似恒等式.我们研究这一概念的一些代数顺从局部紧群。证明了每个$Δ$ -弱$ϕ$ -顺从Banach代数都有一个有界$Δ$ -弱近似恒等式.
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.
Amenability of ultraproducts of Banach algebras
研究了巴拿赫代数的某些性质在超幂构造下何时稳定。特别地，我们考虑了当$\mc A$的每一个超幂都是Arens正则的，并给出了一些证据，证明了这是当且仅当$\mc A$同构于超自反巴拿赫空间上的一个闭算子子代数.我们表明，这样的想法是密切相关的，是否可以合理地定义一个对偶巴拿赫代数的超幂。我们研究了超幂的张量积的行为，并应用它来研究$\mc A$的每一个超幂何时是顺从的问题。我们提供了一个抽象的表征方面的东西，如一个近似的对角线，并考虑当每一个超幂的C $^*$ -代数，或一组$L^1$ -卷积代数，是顺从的。
Amenability for dual Banach algebras
我们定义一个巴拿赫代数A是对偶的，如果$A = (A_\ast)^\ast$对于$A^\ast$的闭子模$A_\ast$.对偶巴拿赫代数类包括所有的$W^\ast$ -代数，也包括局部紧群G的所有代数M（G），自反Banach空间E的所有代数L（E），以及Arens正则巴拿赫代数的所有双链。一般的印象是，顺从，对偶巴拿赫代数，而不是规则的例外。我们证实了这一印象。我们首先证明了在一定条件下，顺从的对偶巴拿赫代数已经是超顺从的，因此是有限维的。然后，我们开发了两个概念的顺从性-康纳斯顺从性和强康纳斯顺从性-考虑$w^\ast$ -拓扑对偶巴拿赫代数。我们将Arens正则巴拿赫代数A的顺从性与$A^{\ast\ast}$的（强）Connes顺从性联系起来;作为应用，我们证明了存在具有逼近性质的自反Banach空间，使得L（E）不是Connes顺从的。利用伪测度代数刻画了内顺从局部紧群的顺从性。最后，我们证明了顺从von Neumann代数是次齐次代数，从而避免了$C^\ast$ -代数的顺从性与核性的等价性.
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. …
On Character Amenability of Banach Algebras
与Banach代数$A$的一个非零同态$φ$相关联，我们考虑了$A^\ast$的某些子空间上的特殊泛函，如$m_φ$，它提供了在$A$的相应极大理想中存在有界右近似恒等式的等价陈述.例如，应用不动点定理产生一个等价的陈述$m_φ$上$A^\ast$的存在，此外，我们阐述的情况下，如果一个功能$m_φ$是唯一的，那么$m_φ$属于拓扑中心的双对偶代数$A^{\ast\ast}$。一个函数代数的例子，令人惊讶的是，矛盾的猜想，一个Banach代数$A$是顺从的，如果$A$是$φ$ -顺从在每个字符$φ$，如果泛函$m_φ$相关的字符$φ$是一致有界的。对两个给定的巴拿赫代数的直和也作了详细的讨论。
Module amenability of the second dual and module topological center of semigroup algebras
设$S$是具有向上幂等元集$E$的逆半群.本文定义了一个巴拿赫代数的第二对偶的模拓扑中心，该代数是另一个巴拿赫代数上的具有相容作用的Banach模，并对$ \ell ^{1}(S)^{**}$（作为$\ell^{1}(E)$ -模）找到了它.我们还证明了$ \ell ^{1}(S)^{**}$是$\ell^{1}(E)$ -模顺从的当且仅当$S$的一个适当的群同态像是有限的。
Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra
如果$D:A \to X$是一个从巴拿赫代数到一个压缩的巴拿赫$A$ -双模的导子，那么可以给$X^{**}$赋予一个$A^{**}$ -双模结构，使得第二个转置$D^{**}: A^{**} \to X^{**}$也是一个导子.我们证明了一个类似的扩展结果，其中$A^{**}$被$\F(A)$，$A$的\emph{enveloping dual Banach algebra}，和$X^{**}$的一个适当的通用的，包络的，正常的对偶双模的$X$。利用这一性质，我们得到了$\F(A)$的连通性的一些新的刻画。特别是，我们表明，$\F(A)$是康纳斯服从当且仅当$A$承认一个所谓的WAP虚拟对角线。我们表明，当$A=L^1(G)$，存在的WAP虚拟对角线是等价的存在的虚拟对角线在通常意义上。我们的方法不涉及$G$不变的手段。
Module Amenability for Semigroup Algebras
本文将巴拿赫代数$A$的顺从性概念推广到$A$上存在额外的$\mathfrak A$ -模结构的情形，并证明了当$S$是一个逆半群且其子半群$E$是幂等元时，则作为$\mathfrak A=\ell^1(E)$上Banach模的$A=\ell^1(S)$是模顺从的当且仅当$S$是顺从的.当$S$是一个离散群时，$\ell^1(E)=\mathbb C$，这就是著名的约翰逊定理。
The pseudo-amenability of Brandt Banach semigroup algebras is considered.
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