topological algebra

In mathematics, a topological algebra $A$ 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 $A$ over a topological field $K$ is a topological vector space together with a bilinear multiplication

:$\cdot: A \times A \to A$,
:$(a,b) \mapsto a \cdot b$

that turns $A$ into an algebra over $K$ 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'': for each neighbourhood of zero $U\subseteq A$ there are neighbourhoods of zero $V\subseteq A$ and $W\subseteq A$ such that $V \cdot W\subseteq U$ (in other words, this condition means that the multiplication is continuous as a map between topological spaces or
* ''stereotype continuity'': for each totally bounded set $S\subseteq A$ and for each neighbourhood of zero $U\subseteq A$ there is a neighbourhood of zero $V\subseteq A$ such that $S \cdot V\subseteq U$ and $V \cdot S\subseteq U$, or
* ''separate continuity'': for each element $a\in A$ and for each neighbourhood of zero $U\subseteq A$ there is a neighbourhood of zero $V\subseteq A$ such that $a\cdot V\subseteq U$ and $V\cdot a\subseteq U$.

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

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References

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Topological vector spaces
Algebras

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