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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 $(A,|| \cdot |{|_A},{e_A})$ and $(B,|| \cdot |{|_B},{e_B})$ are commutative, semisimple Banach algebras with units and natural norms. Assume $T$ is a linear isometry from $(A,|| \cdot |{|_A})$ onto $(B,|| \cdot |{|_B})$ 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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Article copyright: © Copyright 1985 American Mathematical Society