Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46J05

Retrieve articles in all journals with MSC: 46J05

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society