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Viewing Version
7
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'topological $*$-algebra'
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| Title of object: |
topological $*$-algebra |
| Canonical Name: |
TopologicalAlgebra |
| Type: |
Definition |
| Created on: |
2004-10-22 07:03:58 |
| Modified on: |
2004-11-04 04:35:29 |
| Classification: |
msc:46K05, msc:16W10, msc:16W80, msc:22A30, msc:46H35 |
| Defines: |
involution $*$-algebra |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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\newenvironment{df}[1][]{\par\noindent\textbf{Definition (#1)}}{} |
Content:
\begin{df}[Involution]
An involution on an algebra $A$ over a field $F$ is a map $\cdot^* : A \to A : a \mapsto a^*$ such that for every $\{a, b\} \subset A$ and $l \in F$ we have
\begin{enumerate}
\item $a^{**} = a$,
\item $(ab)^* = b^* a^*$ and
\item $(la+b)^* = l^*a^* + b^*$.
\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} |
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