Isometries in semisimple, commutative Banach algebras

Author:
Krzysztof Jarosz

Journal:
Proc. Amer. Math. Soc. **94** (1985), 65-71

MSC:
Primary 46J05

DOI:
https://doi.org/10.1090/S0002-9939-1985-0781058-9

MathSciNet review:
781058

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Abstract: We show that for any semisimple, commutative, complex Banach algebra with unit there are norms on , which we call natural norms, equivalent to the original norm on with the following property: Let and are commutative, semisimple Banach algebras with units and natural norms. Assume is a linear isometry from onto with . Then is an isomorphism in the category of Banach algebras. For a fairly large class of algebras, for example, for uniform algebras, for algebras of the form , the natural norm we have defined coincides with a usual norm.

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0781058-9

Article copyright:
© Copyright 1985
American Mathematical Society