topological $*$-algebra

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

1. $a^{**}=a$,

2. ${(ab)}^{*}=b^{*}{a}^{*}$ and

3. ${(\lambda a+b)}^{*}={\lambda }^{*}{a}^{*}+b^{*}$, where ${\lambda }^{*}$ denotes the involution (http://planetmath.org/InvolutaryRing) of $\lambda$ in $F$.

Definition ($\mathrm{*}$-Algebra) A $*$-algebra is an algebra with an involution.

Definition (Topological $\mathrm{*}$-algebra) A topological $*$-algebra is a $*$-algebra which is also a topological vector space such that its algebra multiplication and involution are continuous.

0.0.1 Remarks:

• $*$-algebras are a particular of involutory rings.

• The involutory field $F$ is often taken as $ℂ$, 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.

Title	topological $*$-algebra
Canonical name	Topologicalalgebra
Date of creation	2013-03-22 14:45:38
Last modified on	2013-03-22 14:45:38
Owner	HkBst (6197)
Last modified by	HkBst (6197)
Numerical id	12
Author	HkBst (6197)
Entry type	Definition
Classification	msc 22A30
Classification	msc 16W80
Classification	msc 16W10
Classification	msc 46K05
Classification	msc 46H35
Synonym	topological *-algebra
Related topic	BanachAlgebra
Related topic	WeakHopfCAlgebra2
Related topic	VonNeumannAlgebra
Defines	involution $*$-algebra
Defines	*-algebra