Topological divisor of zero: Difference between revisions
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In [[Mathematics|mathematics]], an element | In [[HandWiki:Mathematics|mathematics]], an element <math>z</math> of a [[Banach algebra]] <math>A</math> is called a '''topological divisor of zero''' if there exists a [[Biology:Sequence|sequence]] <math>x_1,x_2,x_3,...</math> of elements of <math>A</math> such that | ||
# The sequence | # The sequence <math>zx_n</math> converges to the zero element, but | ||
# The sequence | # The sequence <math>x_n</math> does not converge to the zero element. | ||
If such a sequence exists, then one may assume that | If such a sequence exists, then one may assume that <math>\left \Vert \ x_n \right \| = 1</math> for all <math>n</math>. | ||
If | If <math>A</math> is not commutative, then <math>z</math> is called a "left" topological divisor of zero, and one may define "right" topological divisors of zero similarly. | ||
==Examples== | ==Examples== | ||
* If | * If <math>A</math> has a unit element, then the invertible elements of <math>A</math> form an [[Open set|open subset]] of <math>A</math>, while the non-invertible elements are the complementary [[Closed set|closed subset]]. Any point on the [[Boundary (topology)|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]]). | * In particular, any quasinilpotent element is a topological divisor of zero (e.g. the [[Volterra operator]]). | ||
* An operator on a Banach space <math>X</math>, which is injective, not surjective, but whose image is dense in <math>X</math>, is a left topological divisor of zero. | * An operator on a Banach space <math>X</math>, which is injective, not surjective, but whose image is dense in <math>X</math>, is a left topological divisor of zero. | ||
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The notion of a topological divisor of zero may be generalized to any [[Topological algebra|topological algebra]]. If the algebra in question is not [[First-countable space|first-countable]], one must substitute [[Net (mathematics)|nets]] for the sequences used in the definition. | The notion of a topological divisor of zero may be generalized to any [[Topological algebra|topological algebra]]. If the algebra in question is not [[First-countable space|first-countable]], one must substitute [[Net (mathematics)|nets]] for the sequences used in the definition. | ||
==References== | |||
{{reflist}} | |||
{{Refbegin}} | |||
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} --> Chapter 10 Exercise 11. | |||
{{Refend}} | |||
{{DEFAULTSORT:Topological Divisor Of Zero}} | {{DEFAULTSORT:Topological Divisor Of Zero}} | ||
[[Category:Topological algebra]] | [[Category:Topological algebra]] | ||
{{Sourceattribution|Topological divisor of zero}} | {{Sourceattribution|Topological divisor of zero}} | ||
Latest revision as of 00:25, 21 July 2025
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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