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 $\u2102$, 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  20130322 14:45:38 
Last modified on  20130322 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 