Hilbert moduleHilbertModule2002-08-30 14:07:492004-04-16 10:59:53Definitionpre-Hilbert module\usepackage{amssymb}
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\newcommand*{\defn}[1]{\textbf{#1}}\begin{definition}
A \textbf{(right) pre-Hilbert module} over a $C^*$-algebra $A$ is a right $A$-module $\hilbmod$
equipped with an $A$-valued inner product $\langle-,-\rangle\colon \hilbmod \times \hilbmod \to A$,
i.e.\ a sesquilinear pairing satisfying
\begin{eqnarray}
\langle u,va\rangle & = & \langle u,v\rangle a \\
\langle u,v\rangle & = & \langle v,u\rangle^* \\
\langle v,v\rangle & \geq & 0, \mbox{ with\ } \langle v,v\rangle = 0 \mbox{ iff\ } v = 0,
\end{eqnarray}
for all $u,v \in \hilbmod$ and $a \in A$.
Note, positive definiteness is well-defined due to the notion of positivity for $C^*$-algebras.
The norm of an element $v \in \hilbmod$ is defined by $\norm{v} = \sqrt{\norm{\langle v,v\rangle}}$.
\end{definition}
\begin{definition}
A \textbf{(right) Hilbert module} over a $C^*$-algebra $A$ is a right pre-Hilbert module over $A$ which is complete with respect to the norm.
\end{definition}
\begin{example}[Hilbert spaces]
A complex Hilbert space is a Hilbert $\Cset$-module.
\end{example}
\begin{example}[$C^*$-algebras]
A $C^*$-algebra $A$ is a Hilbert $A$-module with inner product
$\langle a,b\rangle = a^*b$.
\end{example}
\begin{definition}
A \textbf{Hilbert $A$-$B$-bimodule} is a (right) Hilbert module $\hilbmod$ over a $C^*$-algebra $B$ together with a *-homomorphism $\pi$ from a $C^*$-algebra $A$ to $\End(\hilbmod)$.
\end{definition}