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.