nLab
Hilbert module

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)

   $$⟨-,-⟩:H\times H\to B$$\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

$$⟨v\cdot b,w⟩=b^{*}\cdot ⟨v,w⟩.$$\langle v \cdot b,w\rangle = b^\ast \cdot \langle v,w\rangle \,.

Because:

$$\begin{array}{rl}⟨v\cdot b,w⟩& =⟨w,v\cdot b{⟩}^{*}\\ & ={(⟨w,v⟩\cdot b)}^{*}\\ & =b^{*}\cdot ⟨w,v{⟩}^{*}\\ & =b^{*}\cdot ⟨v,w⟩\end{array}.$$\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 $C_0(X)$ 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 ${\Gamma }_0(E)$ of continuous compactly supported sections is a Hilbert $C^{*}$-module over $C_0(X)$ with $C_0(X)$-valued inner product $⟨-,-⟩$ the pointwise inner product in the Hilbert space fiber of $E$:

$$⟨{\sigma }_1,{\sigma }_2⟩(x)≔⟨{\sigma }_1(y),{\sigma }_2(y){⟩}_{E_y}\in C_0(X),{\sigma }_1,{\sigma }_2\in \Gamma (E),x\in X.$$\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 $C_0(X)$-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

$$⟨a_1,a_2⟩≔a_1^{*}\cdot a\in A$$\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 $\{a_n\in A{\}}_{n\in \mathbb{N}}$ 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

$$⟨\{a_n\},\{b_n\}⟩≔\sum _n{a}_n^{*}{b}_n$$\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 $U(\mathscr{H})$ of an infinite-dimensional separable Hilbert space $\mathscr{H}$ is contractible (by Kuiper's theorem), hence so is the classifying space, and so unitary $\mathscr{H}$-fiber bundles (over actual topological spaces) all trivializable. Since moreover $\mathscr{H}\simeq {\ell }^2(ℂ)$ the Hilbert module of example 2 for the trivial $\mathscr{H}$-bundle over $C_0(X)$ is equivalent to ${\ell }^2(C_0(X))$. 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

$$⟨F-,-⟩=⟨-,F^{*}-⟩.$$\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

$$T:v\mapsto \sum _{i=1}^n{w}_i⟨v_i,v⟩$$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 $𝒦(H_1,H_2)$ 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

Related concepts

• Hilbert bimodule

• Mishchenko-Fomenko index theorem

References

• wikipedia, Hilbert $C^{*}$-module