Absolute continuity in the dual of a Banach algebra

Abstract

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