|
|
|
Viewing Version
8
of
'topological $*$-algebra'
|
[ view 'topological $*$-algebra'
|
back to history
]
| Title of object: |
topological $*$-algebra |
| Canonical Name: |
TopologicalAlgebra |
| Type: |
Definition |
| Created on: |
2004-10-22 07:03:58 |
| Modified on: |
2007-08-29 08:00:28 |
| Classification: |
msc:46K05, msc:16W10, msc:16W80, msc:22A30, msc:46H35 |
| Defines: |
involution $*$-algebra, *-algebra |
| Synonyms: |
topological $*$-algebra=topological *-algebra |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newenvironment{df}[1][]{\par\noindent\textbf{Definition (#1)}}{} |
Content:
\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}
|
|
|
|
|
|
