Approximate amenability for Banach algebras

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A Banach algebra is an abstract mathematical object that has both an algebraic and a topological structure, and mathematicians are eager to study the relationship of these two structures. They do this because the abstract objects seem to be of beauty and interest in their own right, but more because these objects cover many naturally occurring examples in mathematics.A very successful line of investigation covers that of amenable Banach algebras, and there are many fine theorems on such algebras. Nevertheless this concept does not cover all appropriate circumstances, and there is now considerable study of variations of this fundamental idea. Some of these variations were introduced by Ghahramani and Loy; essentially, they now consider approximately amenable Banach algebras .I propose to work with Dr. R. J. Loy (Australian National University, Canberra) to study these variants and determine which specfic Banach algebras possess the relevant properties, with a view to formulating a general theory.