topological $*$-algebraTopologicalAlgebra2004-10-22 07:03:582007-08-29 08:01:50Definitioninvolution $*$-algebra*-algebra% this is the default PlanetMath preamble. as your knowledge
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\newenvironment{df}[1][]{\par\noindent\textbf{Definition (#1)}}{}\PMlinkescapeword{involution}
\begin{df}[Involution]
An involution on an algebra $A$ over an \PMlinkname{involutory field}{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
\begin{enumerate}
\item $a^{**} = a$,
\item $(ab)^* = b^* a^*$ and
\item $(\lambda a+b)^* = \lambda^*a^* + b^*$, where $\lambda^*$ denotes the \PMlinkname{involution}{InvolutaryRing} of $\lambda$ in $F$.
\end{enumerate}
\end{df}
\begin{df}[$*$-Algebra]
A $*$-algebra is an algebra with an involution.
\end{df}
\begin{df}[Topological $*$-algebra]
A topological $*$-algebra is a $*$-algebra which is also a topological vector space such that its algebra multiplication and involution are continuous.
\end{df}
\subsubsection{Remarks:}
\begin{itemize}
\item $*$-algebras are a particular \PMlinkescapetext{type} of involutory rings.
\item The involutory field $F$ is often taken as $\mathbb{C}$, where the involution is given by complex conjugation. In this case, condition 3 could be rewritten as:
3.$\;(\lambda a +b)^*= \overline{\lambda}a^*+b^*$
\item Banach *-algebras are topological $*$-algebras.
\end{itemize}