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Perturbations of Uniform Algebras

Perturbations of Uniform Algebras KRZYSZTOF JAROSZ 1. Introduction. Let A be a Banach algebra. By an £-perturbation of A we mean a commutative associative multiplication x on the Banach space A such that for all / , g in A. A Banach algebra is said to be rigid if any sufficiently small perturbation produces a new algebra which is algebraically isomorphic to the original one. Conditions which are sufficient to ensure that a Banach algebra is rigid have been given by B. E. Johnson [1] and independently by I. Raeburn and J. L. Taylor [3]. R. Rochberg [4] considered a wider class of algebras and has proved that if A is a uniform algebra such that its Shilov boundary is equal to its Choquet boundary and each point of the Shilov boundary is G then the Shilov boundaries of A and A 6 x are homeomorphic for any sufficiently small perturbation. In this paper we prove that the assumption "each point of the Shilov boundary is G " can be omitted. As a consequence we obtain a proof that the algebra C(S) is rigid for any compact Hausdorff space S. Before passing to the theorem let us formulate some general facts about e-perturbations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.2.133
Publisher site
See Article on Publisher Site

Abstract

KRZYSZTOF JAROSZ 1. Introduction. Let A be a Banach algebra. By an £-perturbation of A we mean a commutative associative multiplication x on the Banach space A such that for all / , g in A. A Banach algebra is said to be rigid if any sufficiently small perturbation produces a new algebra which is algebraically isomorphic to the original one. Conditions which are sufficient to ensure that a Banach algebra is rigid have been given by B. E. Johnson [1] and independently by I. Raeburn and J. L. Taylor [3]. R. Rochberg [4] considered a wider class of algebras and has proved that if A is a uniform algebra such that its Shilov boundary is equal to its Choquet boundary and each point of the Shilov boundary is G then the Shilov boundaries of A and A 6 x are homeomorphic for any sufficiently small perturbation. In this paper we prove that the assumption "each point of the Shilov boundary is G " can be omitted. As a consequence we obtain a proof that the algebra C(S) is rigid for any compact Hausdorff space S. Before passing to the theorem let us formulate some general facts about e-perturbations.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1983

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