Topological algebra: Difference between revisions
I added the definition of stereotype continuous multiplication |
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:<math>(a,b)\longmapsto a\cdot b</math> |
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that turns <math>A</math> into an [[algebra over a field|algebra]] over <math>K</math> and is continuous in |
that turns <math>A</math> into an [[algebra over a field|algebra]] over <math>K</math> and is continuous in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements (ordered according to the successive weakening of requirements): |
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* ''joint continuity''{{sfn|Beckenstein|Narici|Suffel|1977}}: for each neighbourhood 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 <math>A\times A \longrightarrow A</math>), or |
* ''joint continuity''{{sfn|Beckenstein|Narici|Suffel|1977}}: for each neighbourhood 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 <math>A\times A \longrightarrow A</math>), or |
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* ''stereotype continuity''{{sfn|Akbarov|2003}}: for each [[totally bounded set#Definitions in other contexts|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\quad \&\quad V\cdot S\subseteq U</math>. |
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* ''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>. |
* ''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>. |
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In the first case <math>A</math> is called a ''topological algebra with jointly continuous multiplication'', and in the |
In the first case <math>A</math> is called a ''topological algebra with jointly continuous multiplication'', and in the third ''- with separately continuous multiplication''. |
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A unital [[associative algebra|associative]] topological algebra is (sometimes) called a [[topological ring]]. |
A unital [[associative algebra|associative]] topological algebra is (sometimes) called a [[topological ring]]. |
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:1. [[Fréchet algebra]]s are examples of associative topological algebras with jointly continuous multiplication. |
:1. [[Fréchet algebra]]s are examples of associative topological algebras with jointly continuous multiplication. |
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:2. [[Banach algebra]]s are special cases of [[Fréchet algebra]]s. |
:2. [[Banach algebra]]s are special cases of [[Fréchet algebra]]s. |
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:3. [[Stereotype algebra]]s are examples of associative topological algebras with |
:3. [[Stereotype algebra]]s are examples of associative topological algebras with stereotype continuous multiplication. |
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== External links == |
== External links == |
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==References== |
==References== |
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* {{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 | ref = harv}} |
* {{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 | ref = harv}} |
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*{{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| ref = harv}} |
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* {{cite book | last=Mallios | first=A. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=1986 | isbn=9780080872353 | ref = harv}} |
* {{cite book | last=Mallios | first=A. | title=Topological Algebras | publisher=North Holland | location=Amsterdam | year=1986 | isbn=9780080872353 | ref = harv}} |
Revision as of 09:18, 27 August 2019
In mathematics, a topological algebra is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra over a topological field is a topological vector space together with a bilinear multiplication
- ,
that turns into an algebra over and is continuous in some definite sense. Usually the continuity of the multiplication is expressed by one of the following (non-equivalent) requirements (ordered according to the successive weakening of requirements):
- joint continuity[1]: for each neighbourhood of zero there are neighbourhoods of zero and such that (in other words, this condition means that the multiplication is continuous as a map between topological spaces ), or
- stereotype continuity[2]: for each totally bounded set and for each neighbourhood of zero there is a neighbourhood of zero such that .
- separate continuity[3]: for each element and for each neighbourhood of zero there is a neighbourhood of zero such that and .
In the first case is called a topological algebra with jointly continuous multiplication, and in the third - with separately continuous multiplication.
A unital associative topological algebra is (sometimes) called a topological ring.
History
The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).
Examples
- 1. Fréchet algebras are examples of associative topological algebras with jointly continuous multiplication.
- 2. Banach algebras are special cases of Fréchet algebras.
- 3. Stereotype algebras are examples of associative topological algebras with stereotype continuous multiplication.
External links
- Topological algebra at the nLab
Notes
References
- Beckenstein, E.; Narici, L.; Suffel, C. (1977). Topological Algebras. Amsterdam: North Holland. ISBN 9780080871356.
{{cite book}}
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- Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133.
{{cite journal}}
: Invalid|ref=harv
(help)
- Mallios, A. (1986). Topological Algebras. Amsterdam: North Holland. ISBN 9780080872353.
{{cite book}}
: Invalid|ref=harv
(help)