Absolute continuity in the dual of a Banach algebra

Abstract:

If A is a Banach algebra, G is in the dual space ${A^{\ast }}$, and I is a closed ideal in A, then let ${\left \| G \right \|_{{I^{\ast }}}}$ denote the norm of the restriction of G to I. We define a relation $\ll$ in ${A^{\ast }}$ as follows: $G \ll L$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that if I is a closed ideal in A and ${\left \| L \right \|_{{I^{\ast }}}} < \delta$ then ${\left \| G \right \|_{{I^{\ast }}}} < \varepsilon$. 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 ${C^1}[0,1]$ 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.