In the present paper, the concepts of module (uniform) approximate amenability and contractibility of Banach algebras that are modules over another Banach algebra, are introduced. The general theory is developed and some hereditary properties are given. In analogy with the Banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same ...

In this paper, we study several problems in Banach algebras concerned with the Arens products.

Approximately finite (AF) C *-algebras are classified by approximately finite (r-discrete principal) groupoids. Certain natural triangular subalgebras of AF C *-algebras are similarly classified by triangular subsemigroupoids of AF groupoids [10]. Putting this in a more intuitive way, such subalgebras A are classified by the topologised fundamental binary relation R(A) induced on the Gelfand sp...

A properly in nite C∗-algebra has no non-zero traces. We construct properly in nite Banach ∗-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly `close' to the class of C∗-algebras, in the sense that they can be hermitian or ∗-semisimple.