Absolute continuity in the dual of a Banach algebra

Author:
Stephen Jay Berman

Journal:
Trans. Amer. Math. Soc. **243** (1978), 169-194

MSC:
Primary 46H05; Secondary 46J05

MathSciNet review:
502901

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Abstract: If *A* is a Banach algebra, *G* is in the dual space , and *I* is a closed ideal in *A*, then let denote the norm of the restriction of *G* to *I*. We define a relation in as follows: if for every there exists a such that if *I* is a closed ideal in *A* and then . 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 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0502901-5

Keywords:
Ideals in Banach algebras,
absolute continuity,
Radon-Nikodym theorems

Article copyright:
© Copyright 1978
American Mathematical Society