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

* 我们建议研究和分类某些Banach代数的性质，所有的连续导子从他们到某些模在这些代数是限制（在各种模式的收敛）的内部导子。 我们把这些性质称为广义的顺从性概念。我们建议在过去工作的基础上再接再厉，努力回答由于我们过去的工作而产生的新问题。我们要研究的一个新出现的概念涉及具有这样的性质的代数，即所有的连续导子--如上所定义的--可以用半内映射来近似（这里术语“半内”用于从代数A到A-双模X的映射D，使得存在X的元素m和n，对于A中的所有a，D（a）= a. m-n. a）。我们称这样的代数为半近似顺从代数。我们打算为这个新概念发展一般理论，并研究与之有关的各种Banach代数。** 我们特别感兴趣的是研究g.n.a.抽象调和分析理论的Banach代数的性质（与局部紧群有关的Banach代数）。局部紧拓扑群是具有局部紧拓扑的群，使得群的乘积是联合连续的，并且群反演也是连续的。每一个局部紧拓扑群G都有一个测度m，定义在包含所有Borel集的G子集的σ-代数上，并且m在非空开集上取正值，除了是左扩张不变的（这被称为群的左Haar测度，并且在常数倍以下是唯一的）。局部紧拓扑群上的权w是一个正值连续函数，也是次乘法的。 一个p-Beurling代数（p=1或p>1），是属于这个空间的所有m-可测函数（等价类）的空间 L^p（G，wdm），在群和/或权上规定了一定的条件，以确保空间中任何两个元素的卷积被定义，并将空间转化为Banach代数。在p=1的情况下，空间在卷积积下自动闭合。本文主要研究p-Beurling代数的导子、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Connes顺从性。局部紧群的傅立叶代数是连续函数的Banach代数的推广，连续函数是L^1（R）中函数的傅立叶变换。我们已经特征的g.n. a性质的傅立叶代数的某些局部紧拓扑群，并打算努力对一个特征g.n. a的傅立叶代数的所有局部紧群。***性质的某些C*-代数，并打算对所有的C*-代数的g.n. a的特征。**

*********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.*********