Topological algebra

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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

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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:

  • 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   and  , or
  • separate continuity:[3] for each element   and for each neighbourhood of zero   there is a neighbourhood of zero   such that   and  .

(Certainly, joint continuity implies stereotype continuity, and stereotype continuity implies separate continuity.) In the first case   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

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The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).

Examples

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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

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References

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  • 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. S2CID 115297067.
  • 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.