Banach algebras in which every element is a Topological zero divisor

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.