A Generalized Fredholm Theory for Certain Maps in the Regular Representations of an Algebra

Abstract

Given an algebra A, the elements of A induce linear operators on A by left and right multiplication. Various authors have studied Banach algebras A with the property that some or all of these multiplication maps are completely continuous operators on A ; see (1-5). In (3)1. Kaplansky defined an element u of a Banach algebra A to be completely continuous if the maps a ⟶ ua and a ⟶ au, a ∊ A, are completely continuous linear operators.