Banach algebras in which every element is a topological zero divisor

Abstract

Every element of a complex Banach algebra ( A , ‖ ⋅ ‖ ) (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 ∂ A \partial A coincides with the Gelfand space Δ ( A ) \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 ∂ A = Δ ( A ) \partial A = \Delta (A) can be omitted, and that (b) in case (ii), ∂ A = Δ ( A ) \partial A = \Delta (A) is not a necessary condition.