# Topological algebra

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:

*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 , or

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

(Certanly, 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

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

## 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.