Topological divisor of zero

In mathematics, an element ${\displaystyle z}$ of a Banach algebra ${\displaystyle A}$ is called a topological divisor of zero if there exists a sequence ${\displaystyle x_{1},x_{2},x_{3},...}$ of elements of ${\displaystyle A}$ such that

1. The sequence ${\displaystyle zx_{n}}$ converges to the zero element, but
2. The sequence ${\displaystyle x_{n}}$ does not converge to the zero element.

If such a sequence exists, then one may assume that ${\displaystyle \left\Vert \ x_{n}\right\|=1}$ for all ${\displaystyle n}$.

If ${\displaystyle A}$ is not commutative, then ${\displaystyle z}$ is called a "left" topological divisor of zero, and one may define "right" topological divisors of zero similarly.

Examples

• If ${\displaystyle A}$ has a unit element, then the invertible elements of ${\displaystyle A}$ form an open subset of ${\displaystyle A}$, 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 ${\displaystyle X}$, which is injective, not surjective, but whose image is dense in ${\displaystyle X}$, 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.

References