PlanetMath: topological $*$-algebra

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topological $*$

-algebra

(Definition)

Definition (Involution) An involution on an algebra over an involutory field is a map $\cdot^* : A \to A : a \mapsto a^*$ such that for every $\{a, b\} \subset A$ and $\lambda \in F$ we have

$a^{**} = a$ ,
and
$(\lambda a+b)^* = \lambda^*a^* + b^*$ , where $\lambda^*$ denotes the involution of $\lambda$ in .

Definition (-Algebra) A -algebra is an algebra with an involution.

Definition (Topological -algebra) A topological -algebra is a -algebra which is also a topological vector space such that its algebra multiplication and involution are continuous.

Remarks:

-algebras are a particular type of involutory rings.
The involutory field is often taken as $\mathbb{C}$ , where the involution is given by complex conjugation. In this case, condition 3 could be rewritten as:
3. $\;(\lambda a +b)^*= \overline{\lambda}a^*+b^*$
Banach *-algebras are topological -algebras.

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"topological $*$

-algebra" is owned by HkBst. [ full author list (2) ]

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See Also: Banach algebra

Other names:

topological *-algebra

Also defines:

involution $*$

-algebra, *-algebra

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Cross-references: Banach *-algebras, complex conjugation, field, involutory rings, continuous, multiplication, topological vector space, map, algebra
There are 5 references to this entry.

This is version 9 of topological $*$

-algebra, born on 2004-10-22, modified 2007-08-29.
Object id is 6402, canonical name is TopologicalAlgebra.
Accessed 2554 times total.

Classification:

AMS MSC:	46K05 (Functional analysis :: Topological algebras with an involution :: General theory of topological algebras with involution)
	16W10 (Associative rings and algebras :: Rings and algebras with additional structure :: Rings with involution; Lie, Jordan and other nonassociative structures)
	16W80 (Associative rings and algebras :: Rings and algebras with additional structure :: Topological and ordered rings and modules)
	22A30 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Other topological algebraic systems and their representations)
	46H35 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: Topological algebras of operators)

Pending Errata and Addenda

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