Banach algebras in which every element is a Topological zero divisor

Bhatt, S. J. ; Dedania, H. V. (1995) Banach algebras in which every element is a Topological zero divisor Proceedings of the American Mathematical Society, 123 (3). pp. 735-737. ISSN 0002-9939

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Official URL: http://www.ams.org/journals/proc/1995-123-03/S0002...

Related URL: http://dx.doi.org/10.1090/S0002-9939-1995-1224613-0

Abstract

Every element of a complex Banach algebra (A, ║ · ║) 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 ∂ A coincides with the Gelfand space Δ(A); and (iii) A is a nonunital hermitian Banach *-algebra with continuous involution. Several algebras of analysis have this property. Examples are discussed to show that (a) neither hermiticity nor ∂ A = Δ (A) can be omitted, and that (b) in case (ii), ∂ A = Δ (A) is not a necessary condition.

Item Type:Article
Source:Copyright of this article belongs to American Mathematical Society.
Keywords:Topological Divisor of Zero; Hermitian Banach-algebra; Orthogonal Basis; Uniform Banach Algebra
ID Code:59691
Deposited On:07 Sep 2011 05:16
Last Modified:18 May 2016 10:10

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