# Topological algebra

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In [mathematics](/source/mathematics), a '''topological algebra''' <math>A</math> is an [algebra](/source/algebra_over_a_field) and at the same time a [topological space](/source/topological_space), where the algebraic and the topological structures are coherent in a specified sense.

==Definition==
A '''topological algebra''' <math>A</math> over a [topological field](/source/topological_field) <math>K</math> is a [topological vector space](/source/topological_vector_space) together with a bilinear multiplication

:<math>\cdot: A \times A \to A</math>, 
:<math>(a,b) \mapsto a \cdot b</math>

that turns <math>A</math> into an [algebra](/source/algebra_over_a_field) over <math>K</math> and is [continuous](/source/continuous_function_(topology)) in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements:

* ''joint continuity'':{{sfn|Beckenstein|Narici|Suffel|1977}} for each [neighbourhood](/source/neighbourhood_(mathematics)) of zero <math>U\subseteq A</math> there are neighbourhoods of zero <math>V\subseteq A</math> and <math>W\subseteq A</math> such that <math>V \cdot W\subseteq U</math> (in other words, this condition means that the multiplication is continuous as a map between topological spaces {{nowrap|<math>A \times A \to A</math>),}} or
* ''stereotype continuity'':{{sfn|Akbarov|2003}} for each [totally bounded set](/source/totally_bounded_set) <math>S\subseteq A</math> and for each neighbourhood of zero <math>U\subseteq A</math> there is a neighbourhood of zero <math>V\subseteq A</math> such that <math>S \cdot V\subseteq U</math> and <math>V \cdot S\subseteq U</math>, or
* ''separate continuity'':{{sfn|Mallios|1986}} for each element <math>a\in A</math> and for each neighbourhood of zero <math>U\subseteq A</math> there is a neighbourhood of zero <math>V\subseteq A</math> such that <math>a\cdot V\subseteq U</math> and <math>V\cdot a\subseteq U</math>.

(Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case <math>A</math> is called a "''topological algebra with jointly continuous multiplication''", and in the last, "''with separately continuous multiplication''".

A unital [associative](/source/associative_algebra) topological algebra is (sometimes) called a [topological ring](/source/topological_ring).

==History==
The term was coined by [David van Dantzig](/source/David_van_Dantzig); it appears in the title of his [doctoral dissertation](/source/Thesis) (1931).

== Examples ==
:1. [Fréchet algebra](/source/Fr%C3%A9chet_algebra)s are examples of associative topological algebras with jointly continuous multiplication.
:2. [Banach algebra](/source/Banach_algebra)s are special cases of [Fréchet algebra](/source/Fr%C3%A9chet_algebra)s.
:3. [Stereotype algebra](/source/Stereotype_algebra)s are examples of associative topological algebras with stereotype continuous multiplication.

==Notes==
{{reflist}}

== External links ==
* {{nlab|id=topological+algebra|title=Topological algebra}}

==References==
* {{cite book | last1=Beckenstein | first1=E. |  last2=Narici | first2=L. | last3=Suffel | first3=C. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=1977 | isbn=9780080871356 }}
*{{cite journal|last=Akbarov|first=S.S.|title=Pontryagin duality in the theory of topological vector spaces and in topological algebra|journal=Journal of Mathematical Sciences|year=2003|volume=113|issue=2|pages=179–349|doi=10.1023/A:1020929201133|s2cid=115297067|doi-access=free}}
* {{cite book | last=Mallios | first=A. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=1986 | isbn=9780080872353 }}
* {{cite book | last=Balachandran | first=V.K. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=2000 | isbn=9780080543086 }}
* {{cite book | last=Fragoulopoulou | first=M. | title=Topological Algebras with Involution | publisher=North Holland | location=Amsterdam | year=2005 | isbn=9780444520258 }}

Category:Topological algebra
Category:Topological vector spaces
Category:Algebras

{{topology-stub}}

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Adapted from the Wikipedia article [Topological algebra](https://en.wikipedia.org/wiki/Topological_algebra) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Topological_algebra?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.