Topological divisor of zero

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

1. The sequence zx_n converges to the zero element, but
2. 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.

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 [math]\displaystyle{ X }[/math], which is injective, not surjective, but whose image is dense in [math]\displaystyle{ X }[/math], is a left topological divisor of zero.

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.