Banach algebras in which every element is a topological zero divisor

Bhatt, S. J.; Dedania, H. V.

doi:10.1090/S0002-9939-1995-1224613-0

Banach algebras in which every element is a topological zero divisor
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by S. J. Bhatt and H. V. Dedania PDF

Proc. Amer. Math. Soc. 123 (1995), 735-737 Request permission

Abstract:

Every element of a complex Banach algebra $(A,\left \| \cdot \right \|)$ 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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