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

*我们建议对某些Banach代数进行研究和分类，这些代数具有这样的性质：它们在这些代数上的所有连续导子都是内导子的极限(在不同的收敛模式下)。我们将这些性质称为广义顺从性概念(G.n.a.)。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出现的新问题。我们要研究的一个新概念涉及具有如下性质的代数：如上所述，所有的连续导子都可以用半内映射来逼近(这里，术语“半内”用于表示从代数A到A-双模X的映射D，使得对于A中的所有a，存在X的元素m和n，D(A)=Am-n.a)。我们称这种代数为半近似从属代数。我们打算发展这一新概念的一般理论，并研究与之相关的各类Banach代数。*我们对研究g.n.a特别感兴趣。抽象调和分析理论的Banach代数(与局部紧群相关的Banach代数)的性质局部紧拓扑群是指具有局部紧拓扑的群，使得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都有一个定义在包含所有Borel集的子集的sigma-代数上的测度m，并且m在非空开集上取正值，此外它还是左平移不变的(这称为群的左Haar测度，直到一个常数倍数是唯一的)。局部紧拓扑群上的权w是正值连续函数，它也是次可乘的。P-Beurling代数(p=1或p&gt；1)是指属于L^p(G，WDM)空间的m-可测函数的所有(等价类)构成的空间，其群和/或权被规定，以确保空间中任意两个元素的卷积被定义，并将该空间转化为Banach代数。在p=1的情况下，空间在卷积乘积下自动闭合。我们建议研究p-Beurling代数的导子、乘子、等距同构和G.N.A性质。我们还刻画了局部紧拓扑群的加权测度代数的Connes顺从性。*局部紧拓扑群的傅里叶代数是连续函数的Banach代数的推广，它是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，并打算刻画所有局部紧群的傅里叶代数的G.N.A性质。*我们已经对g.n.a进行了描述。某些C*-代数的性质，并试图刻画所有C*-代数的g.n.a。

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