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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Isometries in semisimple, commutative Banach algebras


Author: Krzysztof Jarosz
Journal: Proc. Amer. Math. Soc. 94 (1985), 65-71
MSC: Primary 46J05
MathSciNet review: 781058
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Abstract: We show that for any semisimple, commutative, complex Banach algebra $ A$ with unit there are norms on $ A$, which we call natural norms, equivalent to the original norm on $ A$ with the following property: Let $ (A,\vert\vert \cdot \vert{\vert _A},{e_A})$ and $ (B,\vert\vert \cdot \vert{\vert _B},{e_B})$ are commutative, semisimple Banach algebras with units and natural norms. Assume $ T$ is a linear isometry from $ (A,\vert\vert \cdot \vert{\vert _A})$ onto $ (B,\vert\vert \cdot \vert{\vert _B})$ with $ T{e_A} = {e_B}$. Then $ T$ 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 $ {C^k}(X),{\text{ Lip}}(X),{\text{ AC}}(X)$, the natural norm we have defined coincides with a usual norm.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0781058-9
PII: S 0002-9939(1985)0781058-9
Article copyright: © Copyright 1985 American Mathematical Society