Absolute Continuity in the Dual of a Banach Algebra

Abstract

We explore this relation (which coincides with absolute continuity of measures when A is the algebra of continuous functions on a compact space) and related concepts in the context of several Banach algebras, particularly the algebra C'[0, 1] of differentiable functions and the algebra of continuous functions on the disc with holomorphic extensions to the interior. We also consider generalizations to noncommutative algebras and Banach modules.