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\section*{Hilbert module}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{operator_algebra}{}\paragraph*{{Operator algebra}}\label{operator_algebra}

[[!include AQFT and operator algebra contents]]

\hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory}

[[!include index theory - contents]]

\hypertarget{hilbert_modules}{}\section*{{Hilbert modules}}\label{hilbert_modules}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{algebras_of_adjointable_operators_on_a_hilbert_module}{$C^\ast$-algebras of adjointable operators on a Hilbert module}\dotfill \pageref*{algebras_of_adjointable_operators_on_a_hilbert_module} \linebreak
\noindent\hyperlink{compact_operators_on_a_hilbert_module}{Compact operators on a Hilbert $C^\ast$-module}\dotfill \pageref*{compact_operators_on_a_hilbert_module} \linebreak
\noindent\hyperlink{fredholm_operators}{Fredholm operators}\dotfill \pageref*{fredholm_operators} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


Hilbert module is an abbreaviation both for a Hilbert $C^\ast$-module (this entry) and the analogous notion of a [[Hilbert Q-module]] (see there), where $Q$ is a [[quantale]] (or a [[locale]], in particular).

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{Hilbert $C^\ast$-module} (or simply \emph{Hilbert module}) is a generalization of the notion of \emph{[[Hilbert space]]} where the algebra of [[complex numbers]] is replaced by a possibly more general [[C\emph{-algebra]] $A$. In particular a Hilbert $A$-module has an [[inner product]] which takes values not in $\mathbb{C}$, but in $A$, and such that [[complex conjugation]] is replaced by the [[star-algebra|star-operation]] in $A$.}

Hilbert $C^\ast$-modules naturally appear as modules over [[groupoid convolution algebras]]. Refined to [[Hilbert C\emph{-bimodules]] they serve as generalized [[homomorphism]] between [[C}-algebras]] in [[noncommutative topology]], and, when further equipped with a left weak [[Fredholm module]] as [[cocycles]] in [[KK-theory]].

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\begin{defn}
\label{HilbertCStarModule}\hypertarget{HilbertCStarModule}{}
For $B \in$ [[C\emph{Alg]], a [[Hilbert C}-module]] over $B$ is

\begin{enumerate}%
\item a [[complex numbers|complex]] [[vector space]] $H$;


\item equipped with an [[action]] of $B$ from the right;


\item equipped with a [[sesquilinear map]] (linear in the second argument)

\begin{displaymath}
\langle -,-\rangle \colon H \times H \to B
\end{displaymath}
(the $B$-valued [[inner product]])



\end{enumerate}
such that

\begin{enumerate}%
\item $\langle -,-\rangle$ behaves like a positive definite inner product over $B$ in that for all $x,y \in H$ and $b \in B$ we have

\begin{enumerate}%
\item $\langle x,y\rangle^\ast = \langle y,x\rangle$


\item $\langle x,x\rangle \geq 0$ (in the sense of [[positive elements]] in $B$)


\item $\langle x,x\rangle = 0$ precisely if $x = 0$;


\item $\langle x,y \cdot b\rangle = \langle x,y \rangle \cdot b$



\end{enumerate}

\item $H$ is [[complete space|complete]] with respect to the [[norm]]

${\Vert x \Vert_H} \coloneqq {\Vert \langle x,x\rangle\Vert_B}^{1/2}$.



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In addition to the explicit $B$-linearity in the second argument under right multiplicatojn

\begin{displaymath}
\langle v, w \cdot b\rangle = \langle v,w\rangle \cdot b
\end{displaymath}
the axioms imply conjugate $B$-linearity in the first argument and under left multiplication

\begin{displaymath}
\langle v \cdot b,w\rangle = b^\ast \cdot \langle v,w\rangle
  \,.
\end{displaymath}
Because:

\begin{displaymath}
\begin{aligned}
    \langle v \cdot b,w\rangle
    & =
    \langle w,  v\cdot b\rangle^\ast
    \\
    & = \left( \left\langle w,v\right\rangle \cdot b\right)^\ast
    \\
    & = b^\ast \cdot \langle w,v\rangle^\ast
    \\
    & = b^\ast \cdot \langle v,w\rangle
  \end{aligned}
  \,.
\end{displaymath}
\end{remark}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

First of all we have:

\begin{example}
\label{}\hypertarget{}{}
An ordinary complex [[Hilbert space]] is a Hilbert $\mathbb{C}$-module.

\end{example}
The archetypical class of examples of Hilbert $C^\ast$-modules for [[commutative C\emph{-algebras]] is the following. The general definition \ref{HilbertCStarModule} may be understood as the generalization of the structure of this example to [[noncommutative topology|non-commutative C}-algebras]]. See also remark \ref{TheTrivialHilbertBundleAndL2} below.

\begin{example}
\label{FromAHilbertSpaceBundle}\hypertarget{FromAHilbertSpaceBundle}{}
Let $X$ be a [[locally compact topological space]] and write $C_0(X)$ for its [[C\emph{-algebra]] of [[continuous functions]] [[vanishing at infinity]].}

Let $E \to X$ be a [[fiber bundle]] of [[Hilbert spaces]] over $X$, hence a canonically [[associated bundle]] to a [[unitary group]]-[[principal bundle]]. Then the space $\Gamma_0(E)$ of continuous compactly supported [[sections]] is a Hilbert $C^\ast$-module over $C_0(X)$ with $C_0(X)$-valued [[inner product]] $\langle -,-\rangle$ the pointwise inner product in the [[Hilbert space]] [[fiber]] of $E$:

\begin{displaymath}
\langle \sigma_1, \sigma_2\rangle(x)
  \coloneqq
  \langle \sigma_1(y), \sigma_2(y)\rangle_{E_y}
  \;\in C_0(X)\,,
  \;\;\;\;\;\;
  \sigma_1, \sigma_2 \in \Gamma(E), \; x \in X
  \,.
\end{displaymath}
\end{example}
\begin{prop}
\label{}\hypertarget{}{}
Every Hilbert $C_0(X)$-module arises, up to [[isomorphism]], as in example \ref{FromAHilbertSpaceBundle}.

\end{prop}
\begin{example}
\label{CAstAlgebraAsModuleOverItself}\hypertarget{CAstAlgebraAsModuleOverItself}{}
Every $C^\ast$-algebra $A$ is a Hilbert $A$-module over itself when equipped by with the $A$-valued inner product given simply by

\begin{displaymath}
\langle a_1,a_2\rangle \coloneqq a_1^\ast \cdot a \;\;\in A
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
In view of the archetypical example \ref{FromAHilbertSpaceBundle}, example \ref{CAstAlgebraAsModuleOverItself} may be interpreted as exhibiting the trivial [[complex line bundle]] over whatever space $A$ is the $C^\ast$-algebra of functions on (an actual [[topological space]] if $A$ is a [[commutative C\emph{-algebra]] or else the [[noncommutative topology]] defined as the formal dual of $A$).}

\end{remark}
\begin{example}
\label{L2}\hypertarget{L2}{}
For $A \in$ [[C\emph{Alg]], let $\ell^2 A$ be the space of those [[sequences]] $\{a_n \in A\}_{n \in \mathbb{N}}$ of elements in $A$ such that the [[series]] $\sum_n a_n^\ast a_n$ [[convergence|converges]]. This is a Hilbert $A$-module when equipped with the degreewise $A$-[[C}-representation]], with the $A$-valued inner product

\begin{displaymath}
\langle \{a_n\}, \{b_n\}\rangle
  \coloneqq
  \sum_n a_n^\ast b_n
\end{displaymath}
and after [[completion]] with under the induced [[norm]].

This $\ell^2 A$ is sometimes called the \textbf{standard Hilbert $A$-module} over $A$.

\end{example}
\begin{remark}
\label{TheTrivialHilbertBundleAndL2}\hypertarget{TheTrivialHilbertBundleAndL2}{}
In view of example \ref{FromAHilbertSpaceBundle} we may think of example \ref{L2} as exhibiting the trivial countably-infinite dimensional [[Hilbert space]] bundle over the space dual to $A$.

This is because the [[unitary group]] $U(\mathcal{H})$ of an infinite-dimensional [[separable Hilbert space]] $\mathcal{H}$ is [[contractible topological space|contractible]] (by [[Kuiper's theorem]]), hence so is the [[classifying space]], and so unitary $\mathcal{H}$-fiber bundles (over actual topological spaces) all trivializable. Since moreover $\mathcal{H} \simeq \ell^2(\mathbb{C})$ the Hilbert module of example \ref{FromAHilbertSpaceBundle} for the trivial $\mathcal{H}$-bundle over $C_0(X)$ is equivalent to $\ell^2(C_0(X))$. Example \ref{L2} generalizes this to arbitrary $C*$-algebras $A$.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{algebras_of_adjointable_operators_on_a_hilbert_module}{}\subsubsection*{{$C^\ast$-algebras of adjointable operators on a Hilbert module}}\label{algebras_of_adjointable_operators_on_a_hilbert_module}

\begin{defn}
\label{AdjointableOperator}\hypertarget{AdjointableOperator}{}
For $A \in$ [[C\emph{Alg]] and $H$ a Hilbert $A$-module, def. \ref{HilbertCStarModule}, a $\mathbb{C}$-[[linear operator]] $F \colon H \to H$ is called \textbf{adjointable} if there is an [[adjoint operator]] $F^\ast \colon H \to H$ with respect to the $A$-valued inner product in the sense that}

\begin{displaymath}
\langle F -, -\rangle = \langle -,F^\ast -\rangle
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The adjointable operators on a Hilbert $A$-module, def. \ref{AdjointableOperator}, form a [[Banach algebra|Banach]] [[star-algebra]].

For $A$ itself regarded as a Hilbert $A$-module as in example \ref{CAstAlgebraAsModuleOverItself}, this is the [[multiplier algebra]] of $A$.

\end{prop}
\hypertarget{compact_operators_on_a_hilbert_module}{}\subsubsection*{{Compact operators on a Hilbert $C^\ast$-module}}\label{compact_operators_on_a_hilbert_module}

\begin{defn}
\label{CompactOperator}\hypertarget{CompactOperator}{}
For $H_1, H_2$ two Hilbert $C^\ast$-modules, an adjointable operator $T \colon H_1 \to H_2$, def. \ref{AdjointableOperator}, is of \textbf{finite rank} if it is of the form

\begin{displaymath}
T \colon v \mapsto \sum_{i = 1}^n w_i \langle v_i, v\rangle
\end{displaymath}
for vectors $v_i \in H_1$ and $w_i \in H_2$. $T$ is called a \textbf{generalized compact operator} if it is in the norm-closure of finite-rank operators.

\end{defn}
Typically one writes $\mathcal{K}(H_1, H_2)$ for the space of generalized complact operators.

\hypertarget{fredholm_operators}{}\subsubsection*{{Fredholm operators}}\label{fredholm_operators}

\begin{defn}
\label{}\hypertarget{}{}
An operator $F \colon H_1 \to H_2$ is called a \textbf{generalized [[Fredholm operator]]} if there exists an operator $S \colon H_2 \to H_1$ (then called a [[parametrix]] for $F$) such that both

$F \circ S - id_{H_2}$ and $S \circ F - id_{H_1}$

are [[compact operators]] according to def. \ref{CompactOperator}.

\end{defn}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item Kasparov's [[KK-theory]] is formulated in terms of Hilbert (bi)modules

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hilbert bimodule]]


\item [[Mishchenko-Fomenko index theorem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Hilbert $C^\ast$-modules were introduced by [[Irving Kaplansky]] in

\begin{itemize}%
\item Irving Kaplansky, \emph{Modules over operator algebras}, Amer. J. Math. \textbf{75} (1953) 839--853

\end{itemize}
Contemporary references are

\begin{itemize}%
\item [[Bruce Blackadar]], \emph{[[K-Theory for Operator Algebras]]}, section 13
\item wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Hilbert_C*-module}{Hilbert $C^\ast$-module}}
\item E. Christopher Lance, \emph{Hilbert $C^\ast$-modules, A toolkit for operator algebraists}, London Math. Soc. Lec. Note Ser. \textbf{210}, Cambridge Univ. Press 1995

\end{itemize}
[[!redirects Hilbert module]] [[!redirects Hilbert modules]] [[!redirects Hilbert C\emph{-module]] [[!redirects Hilbert C}-modules]] [[!redirects Hilbert C-star-module]] [[!redirects Hilbert C-star-modules]]



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