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

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 $\left\Vert \ x_{n}\right\|=1$ 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 $X$, 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.

References

• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Chapter 10 Exercise 11.