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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Banach algebras in which every element is a topological zero divisor

Authors: S. J. Bhatt and H. V. Dedania
Journal: Proc. Amer. Math. Soc. 123 (1995), 735-737
MSC: Primary 46H05; Secondary 46K05
MathSciNet review: 1224613
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Abstract: Every element of a complex Banach algebra $ (A,\left\Vert \cdot \right\Vert)$ is a topological divisor of zero, if at least one of the following holds: (i) A is infinite dimensional and admits an orthogonal basis, (ii) A is a nonunital uniform Banach algebra in which the Silov boundary $ \partial A$ coincides with the Gelfand space $ \Delta (A)$; and (iii) A is a nonunital hermitian Banach $ \ast $-algebra with continuous involution. Several algebras of analysis have this property. Examples are discussed to show that (a) neither hermiticity nor $ \partial A = \Delta (A)$ can be omitted, and that (b) in case (ii), $ \partial A = \Delta (A)$ is not a necessary condition.

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Keywords: Topological divisor of zero, hermitian Banach $ \ast $-algebra, orthogonal basis, uniform Banach algebra
Article copyright: © Copyright 1995 American Mathematical Society