Topological divisor of zero

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In mathematics, an element z of a Banach algebra A is called a topological divisor of zero if there exists a sequence x₁, x₂, x₃, ... of elements of A such that

The sequence zx_n converges to the zero element, but
The sequence x_n does not converge to the zero element.

If such a sequence exists, then one may assume that ||x_n|| = 1 for all n.

If A is not commutative, then z is called a left topological divisor of zero, and one may define right topological divisors of zero similarly.

[edit] Examples

If A has a unit element, then the invertible elements of A form an open subset of A, while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero.
In particular, any quasinilpotent element is a topological divisor of zero (e.g. the Volterra operator).
An operator on a Banach space $X$ , which is injective, not surjective, but whose image is dense in $X$ , is a left topological divisor of zero.

[edit] Generalization

The notion of a topological divisor of zero may be generalized to any topological algebra. If the algebra in question is not first-countable, one must substitute nets for the sequences used in the definition.