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.