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

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