# nLab Hilbert module

## Theorems

### Euclidean QFT

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

## Definitions

operator K-theory

K-homology

# Hilbert modules

## Idea

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

Hilbert ${C}^{*}$-modules naturally appear as modules over groupoid convolution algebras. Refined to Hilbert C*-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.

## Definition

###### Definition

For $B\in$ C*Alg, a Hilbert C*-module over $B$ is

1. a complex vector space $H$;

2. equipped with an action of $B$ from the right;

3. equipped with a sesquilinear map (linear in the second argument)

$⟨-,-⟩:H×H\to B$\langle -,-\rangle \colon H \times H \to B

(the $B$-valued inner product)

such that

1. $⟨-,-⟩$ behaves like a positive definite inner product over $B$ in that for all $x,y\in H$ and $b\in B$ we have

1. $⟨x,y{⟩}^{*}=⟨y,x⟩$

2. $⟨x,x⟩\ge 0$ (in the sense of positive elements in $B$)

3. $⟨x,x⟩=0$ precisely if $x=0$;

4. $⟨x,y\cdot b⟩=⟨x,y⟩\cdot b$

2. $H$ is complete with respect to the norm

$\parallel x{\parallel }_{H}≔\parallel ⟨x,x⟩{\parallel }_{B}$.

###### Remark

In addition to the explicit $B$-linearity in the second argument under right multiplicatojn

$⟨v,w\cdot b⟩=⟨v,w⟩\cdot b$\langle v, w \cdot b\rangle = \langle v,w\rangle \cdot b

the axioms imply conjugate $B$-linearity in the first argument and under left multiplication

$⟨v\cdot b,w⟩={b}^{*}\cdot ⟨v,w⟩\phantom{\rule{thinmathspace}{0ex}}.$\langle v \cdot b,w\rangle = b^\ast \cdot \langle v,w\rangle \,.

Because:

$\begin{array}{rl}⟨v\cdot b,w⟩& =⟨w,v\cdot b{⟩}^{*}\\ & ={\left(⟨w,v⟩\cdot b\right)}^{*}\\ & ={b}^{*}\cdot ⟨w,v{⟩}^{*}\\ & ={b}^{*}\cdot ⟨v,w⟩\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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} \,.

## Examples

First of all we have:

###### Example

An ordinary complex Hilbert space is a Hilbert $ℂ$-module.

The archetypical class of examples of Hilbert ${C}^{*}$-modules for commutative C*-algebras is the following. The general definition 1 may be understood as the generalization of the structure of this example to non-cmmutative C*-algebras. See also remark 3 below.

###### Example

Let $X$ be a locally compact topological space and write ${C}_{0}\left(X\right)$ for its C*-algebra of continuous functions of compact support.

Let $E\to X$ be a fiber bundle of Hilbert spaces over $X$, hence an canonically associated bundle to a unitary group-principal bundle. Then the space ${\Gamma }_{0}\left(E\right)$ of continuous compactly supported sections is a Hilbert ${C}^{*}$-module over ${C}_{0}\left(X\right)$ with ${C}_{0}\left(X\right)$-valued inner product $⟨-,-⟩$ the pointwise inner product in the Hilbert space fiber of $E$:

$⟨{\sigma }_{1},{\sigma }_{2}⟩\left(x\right)≔⟨{\sigma }_{1}\left(y\right),{\sigma }_{2}\left(y\right){⟩}_{{E}_{y}}\phantom{\rule{thickmathspace}{0ex}}\in {C}_{0}\left(X\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\sigma }_{1},{\sigma }_{2}\in \Gamma \left(E\right),\phantom{\rule{thickmathspace}{0ex}}x\in X\phantom{\rule{thinmathspace}{0ex}}.$\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 \,.
###### Proposition

Every Hilbert ${C}_{0}\left(X\right)$-module arises, up to isomorphism, as in example 2.

###### Example

Every ${C}^{*}$-algebra $A$ is a Hilbert $A$-module over itself when equipped by with the $A$-valued inner product given simply by

$⟨{a}_{1},{a}_{2}⟩≔{a}_{1}^{*}\cdot a\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in A$\langle a_1,a_2\rangle \coloneqq a_1^\ast \cdot a \;\;\in A
###### Remark

In view of the archetypical example 2, example 3 may be interpreted as exhibiting the trivial complex line bundle over whatever space $A$ is the ${C}^{*}$-algebra of functions on (an actual topological space if $A$ is a commutative C*-algebra or else the noncommutative topology defined as the formal dual of $A$).

###### Example

For $A\in$ C*Alg, let ${\ell }^{2}A$ be the space of those sequences $\left\{{a}_{n}\in A{\right\}}_{n\in ℕ}$ of elements in $A$ such that the series ${\sum }_{n}{a}_{n}^{*}{a}_{n}$ converges. This is a Hilbert $A$-module when equipped with the degreewise $A$-C*-representation, with the $A$-valued inner product

$⟨\left\{{a}_{n}\right\},\left\{{b}_{n}\right\}⟩≔\sum _{n}{a}_{n}^{*}{b}_{n}$\langle \{a_n\}, \{b_n\}\rangle \coloneqq \sum_n a_n^\ast b_n

and after completion with under the induced norm.

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

###### Remark

In view of example 2 we may think of example 4 as exhibiting the trivial countably-infinite dimensional Hilbert space bundle over the space dual to $A$.

This is because the unitary group $U\left(ℋ\right)$ of an infinite-dimensional separable Hilbert space $ℋ$ is contractible (by Kuiper's theorem), hence so is the classifying space, and so unitary $ℋ$-fiber bundles (over actual topological spaces) all trivializable. Since moreover $ℋ\simeq {\ell }^{2}\left(ℂ\right)$ the Hilbert module of example 2 for the trivial $ℋ$-bundle over ${C}_{0}\left(X\right)$ is equivalent to ${\ell }^{2}\left({C}_{0}\left(X\right)\right)$. Example 4 generalizes this to arbitrary $C*$-algebras $A$.

## Properties

### ${C}^{*}$-algebras of adjointable operators on a Hilbert module

###### Definition

For $A\in$ C*Alg and $H$ a Hilbert $A$-module, def. 1, a $ℂ$-linear operator $F:H\to H$ is called adjointable if there is an adjoint operator ${F}^{*}:H\to H$ with respect to the $A$-valued inner product in the sense that

$⟨F-,-⟩=⟨-,{F}^{*}-⟩\phantom{\rule{thinmathspace}{0ex}}.$\langle F -, -\rangle = \langle -,F^\ast -\rangle \,.
###### Proposition

The adjointable operators on a Hilbert $A$-module, def. 2, form a Banach star-algebra.

For $A$ itself regarded as a Hilbert $A$-module as in example 3, this is the multiplier algebra? of $A$.

### Compact operators on a Hilbert ${C}^{*}$-module

###### Definition

For ${H}_{1},{H}_{2}$ two Hilbert ${C}^{*}$-modules, an adjointable operator $T:{H}_{1}\to {H}_{2}$, def. 2, is of finite rank if it is of the form

$T:v↦\sum _{i=1}^{n}{w}_{i}⟨{v}_{i},v⟩$T \colon v \mapsto \sum_{i = 1}^n w_i \langle v_i, v\rangle

for vectors ${v}_{i}\in {H}_{1}$ and ${w}_{i}\in {H}_{2}$. $T$ is called a generalized compact operator if it is in the norm-closure of finite-rank operators.

Typically one writes $𝒦\left({H}_{1},{H}_{2}\right)$ for the space of generalized complact operators.

### Fredholm operators

###### Definition

An operator $F:{H}_{1}\to {H}_{2}$ is called a generalized Fredholm operator if there exists an operator $S:{H}_{2}\to {H}_{1}$ (then called a parametrix for $F$) such that both

$F\circ S-{\mathrm{id}}_{{H}_{2}}$ and $S\circ F-{\mathrm{id}}_{{H}_{1}}$

are compact operators according to def. 3.

## Applications

• Kasparov’s KK-theory is formulated in terms of Hilbert (bi)modules

## References

Revised on May 20, 2013 12:44:22 by Urs Schreiber (89.204.130.66)