Topological divisor of zero
In mathematics, an element [math]\displaystyle{ z }[/math] of a Banach algebra [math]\displaystyle{ A }[/math] is called a topological divisor of zero if there exists a sequence [math]\displaystyle{ x_1,x_2,x_3,... }[/math] of elements of [math]\displaystyle{ A }[/math] such that
- The sequence [math]\displaystyle{ zx_n }[/math] converges to the zero element, but
- The sequence [math]\displaystyle{ x_n }[/math] does not converge to the zero element.
If such a sequence exists, then one may assume that [math]\displaystyle{ \left \Vert \ x_n \right \| = 1 }[/math] for all [math]\displaystyle{ n }[/math].
If [math]\displaystyle{ A }[/math] is not commutative, then [math]\displaystyle{ z }[/math] is called a "left" topological divisor of zero, and one may define "right" topological divisors of zero similarly.
Examples
- If [math]\displaystyle{ A }[/math] has a unit element, then the invertible elements of [math]\displaystyle{ A }[/math] form an open subset of [math]\displaystyle{ A }[/math], 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.
References
- Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. Chapter 10 Exercise 11.
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