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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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 $ {A^{\ast}}$, and I is a closed ideal in A, then let $ {\left\Vert G \right\Vert _{{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\Vert L \right\Vert _{{I^{\ast}}}} < \delta $ then $ {\left\Vert G \right\Vert _{{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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0502901-5
PII: S 0002-9947(1978)0502901-5
Keywords: Ideals in Banach algebras, absolute continuity, Radon-Nikodym theorems
Article copyright: © Copyright 1978 American Mathematical Society