BANACH-ALGEBRAS IN WHICH EVERY ELEMENT IS A TOPOLOGICAL ZERO DIVISOR

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