Topological algebra
In mathematics, a topological algebra [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ A }[/math] over a topological field [math]\displaystyle{ K }[/math] is a topological vector space together with a bilinear multiplication
- [math]\displaystyle{ \cdot: A \times A \to A }[/math],
- [math]\displaystyle{ (a,b) \mapsto a \cdot b }[/math]
that turns [math]\displaystyle{ A }[/math] into an algebra over [math]\displaystyle{ 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:
- joint continuity:[1] for each neighbourhood of zero [math]\displaystyle{ U\subseteq A }[/math] there are neighbourhoods of zero [math]\displaystyle{ V\subseteq A }[/math] and [math]\displaystyle{ W\subseteq A }[/math] such that [math]\displaystyle{ V \cdot W\subseteq U }[/math] (in other words, this condition means that the multiplication is continuous as a map between topological spaces [math]\displaystyle{ A \times A \to A }[/math]), or
- stereotype continuity:[2] for each totally bounded set [math]\displaystyle{ S\subseteq A }[/math] and for each neighbourhood of zero [math]\displaystyle{ U\subseteq A }[/math] there is a neighbourhood of zero [math]\displaystyle{ V\subseteq A }[/math] such that [math]\displaystyle{ S \cdot V\subseteq U }[/math] and [math]\displaystyle{ V \cdot S\subseteq U }[/math], or
- separate continuity:[3] for each element [math]\displaystyle{ a\in A }[/math] and for each neighbourhood of zero [math]\displaystyle{ U\subseteq A }[/math] there is a neighbourhood of zero [math]\displaystyle{ V\subseteq A }[/math] such that [math]\displaystyle{ a\cdot V\subseteq U }[/math] and [math]\displaystyle{ V\cdot a\subseteq U }[/math].
(Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case [math]\displaystyle{ A }[/math] is called a "topological algebra with jointly continuous multiplication", and in the last, "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.
Notes
External links
References
- Beckenstein, E.; Narici, L.; Suffel, C. (1977). Topological Algebras. Amsterdam: North Holland. ISBN 9780080871356.
- 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.
- Mallios, A. (1986). Topological Algebras. Amsterdam: North Holland. ISBN 9780080872353.
- Balachandran, V.K. (2000). Topological Algebras. Amsterdam: North Holland. ISBN 9780080543086.
- Fragoulopoulou, M. (2005). Topological Algebras with Involution. Amsterdam: North Holland. ISBN 9780444520258.
![]() | Original source: https://en.wikipedia.org/wiki/Topological algebra.
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