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 $$x_1,x_2,x_3,...$$ of elements of $$A$$ such that If such a sequence exists, then one may assume that $$\left \Vert \ x_n \right \| = 1$$ for all $$n$$.
 * 1) The sequence $$zx_n$$ converges to the zero element, but
 * 2) The sequence $$x_n$$ does not converge to the zero element.

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 $$X$$, which is injective, not surjective, but whose image is dense in $$X$$, 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.