Generalized notions of amenability and derivations on Banach algebras related to locally compact groups

*********我们提出研究和分类某些Banach代数，这些代数上所有由它们连续导出的模都是内导的极限（在各种收敛模式下）。我们称这些性质为可顺从性的广义概念（g.n.a）。我们建议以我们过去的工作为基础，努力回答由于我们过去的工作而产生的新问题。我们将要研究的一个新出现的概念涉及到具有以下性质的代数：所有的连续导数——如上所定义——都可以用半内映射来近似(这里的“半内”一词用于将D从代数A映射到A-双模X，使得X的元素m和n存在，D(A) = a.m. -n。a，对于所有a)。我们称这样的代数为半近似可服从代数。我们打算为这个新概念发展一般理论，并研究关于它的各种巴拿赫代数。******我们特别感兴趣的是研究抽象调和分析理论中的Banach代数的g.n.a.性质（与局部紧群相关的Banach代数）。局部紧拓扑群是具有局部紧拓扑的群，群的乘积是联合连续的，群的逆也是连续的。每个局部紧拓扑群G承认一个测度m定义在G的子集的sigma代数上，其中G的子集包含所有Borel集合，并且m在非空开集中取正值，除了是左平移不变的（这被称为群的左哈尔测度，直到一个常数倍是唯一的）。局部紧拓扑群上的权值w是一个具有次乘性的正数值连续函数。p- beurling代数（p=1或p>1）是属于空间L^p（G, wdm）的所有m可测函数（等价类）的空间，在群和/或权上规定了一定的条件，以确保空间中任意两个元素的卷积被定义，并将空间转化为巴纳赫代数。当p=1时，空间在卷积积下自动闭合。我们提出研究p-Beurling代数的导数、乘子、等距同构和g.n.a性质。我们还打算刻画局部紧拓扑群的加权测度代数的cones可调性。局部紧群的傅里叶代数是连续函数的Banach代数的推广，连续函数是L^1(R)中函数的傅里叶变换。我们已经刻画了某些局部紧拓扑群的傅里叶代数的g.n.a性质，并打算研究所有局部紧群的傅里叶代数的表征g.n.a。***我们已经表征了某些C*-代数的g.n.a性质，并打算对所有C*-代数的g.n.a进行表征。*********

*********We propose to study and classify certain Banach algebras having the property that all the continuous derivations from them into certain modules over these algebras are limits (in various modes of convergence) of inner derivations. We have called these properties generalized notions of amenability (g.n.a.). We propose to build on our past work and endeavour to answer the new questions that have risen as a result of our past work. A newly emerging notion to be studied by us concerns algebras having the property that all the continuous derivations -- defined as above -- can be approximated by semi-inner mappings (here the term "semi-inner" is used for a mapping D from an algebra A into an A-bimodule X such that there exist elements m and n of X for which, D(a) = a.m -n.a, for all a in A). We call such algebras semi-approximately amenable. We intend to develop general theory for this new notion and investigate various classes of Banach algebras with regard to it. ******We are particularly interested in studying the g.n.a. properties of the Banach algebras of the theory of abstract harmonic analysis (Banach algebras related to locally compact groups). A locally compact topological group is a group with a locally compact topology such the product of the group is jointly continuous and group-inversion is also continuous. Every locally compact topological group G admits a measure m defined on a sigma-algebra of subsets of G containing all the Borel sets and m takes positive values on non-empty open sets, in addition to being left translation-invariant (this is called the left Haar measure of the group and is unique up to a constant multiple) . A weight w on a locally compact topological group is a positive-valued continuous function that is also submultiplicative. A p-Beurling algebra (p=1 or p>1), is the space of all (equivalence classes) of m-measurable functions that belong to the space L^p(G , wdm), with certain conditions stipulated on the group and/or the weight that insure the convolution of any two elements of the space is defined and turns the space into a Banach algebra. In the case p=1, the space is automatically closed under convolution product. We propose to study derivations, multipliers, isometric isomorphisms, and g.n.a properties of p-Beurling algebras. We also intend to characterize the Connes amenability of the weighted measure algebras of locally compact topological groups.***The Fourier algebra of a locally compact group is a generalization of the Banach algebra of continuous functions that are the Fourier transforms of functions in L^1(R). We have already characterized the g.n.a properties of the Fourier algebras of certain locally compact topological groups and intend to work towards a characterization g.n.a for Fourier algebras of all locally compact groups. ***We have already characterized the g.n.a. properties of certain C*-algebras, and intend to work towards characterization of g.n.a for all the C*-algebras.*********