Hilbert modules

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).

Idea

The notion of Hilbert $C^\ast$-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 $\mathbb{C}$, but in $A$, and such that complex conjugation is replaced by the star-operation in $A$.

Hilbert $C^\ast$-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)

$\langle -,-\rangle \colon H \times H \to B$

(the $B$-valued inner product)

such that

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

1. $\langle x,y\rangle^\ast = \langle y,x\rangle$

2. $\langle x,x\rangle \geq 0$ (in the sense of positive elements in $B$)

3. $\langle x,x\rangle = 0$ precisely if $x = 0$;

4. $\langle x,y \cdot b\rangle = \langle x,y \rangle \cdot b$

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

${\Vert x \Vert_H} \coloneqq {\Vert \langle x,x\rangle\Vert_B}$.

Remark

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

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

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

$\langle v \cdot b,w\rangle = b^\ast \cdot \langle v,w\rangle \,.$

Because:

\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 $\mathbb{C}$-module.

The archetypical class of examples of Hilbert $C^\ast$-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-commutative 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 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 $E$:

$\langle \sigma_1, \sigma_2\rangle(x) \coloneqq \langle \sigma_1(y), \sigma_2(y)\rangle_{E_y} \;\in C_0(X)\,, \;\;\;\;\;\; \sigma_1, \sigma_2 \in \Gamma(E), \; x \in X \,.$
Proposition

Every Hilbert $C_0(X)$-module arises, up to isomorphism, as in example 2.

Example

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

$\langle a_1,a_2\rangle \coloneqq a_1^\ast \cdot a \;\;\in A$
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 $C^\ast$-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 $\sum_n a_n^\ast a_n$ converges. This is a Hilbert $A$-module when equipped with the degreewise $A$-C*-representation, with the $A$-valued inner product

$\langle \{a_n\}, \{b_n\}\rangle \coloneqq \sum_n a_n^\ast b_n$

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 $U(\mathcal{H})$ of an infinite-dimensional separable Hilbert space $\mathcal{H}$ is 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 2 for the trivial $\mathcal{H}$-bundle over $C_0(X)$ is equivalent to $\ell^2(C_0(X))$. Example 4 generalizes this to arbitrary $C*$-algebras $A$.

Properties

$C^\ast$-algebras of adjointable operators on a Hilbert module

Definition

For $A \in$ C*Alg and $H$ a Hilbert $A$-module, def. 1, a $\mathbb{C}$-linear operator $F \colon H \to H$ is called 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

$\langle F -, -\rangle = \langle -,F^\ast -\rangle \,.$
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 $C^\ast$-module

Definition

For $H_1, H_2$ two Hilbert $C^\ast$-modules, an adjointable operator $T \colon H_1 \to H_2$, def. 2, is of finite rank if it is of the form

$T \colon v \mapsto \sum_{i = 1}^n w_i \langle v_i, v\rangle$

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 $\mathcal{K}(H_1, H_2)$ for the space of generalized complact operators.

Fredholm operators

Definition

An operator $F \colon H_1 \to H_2$ is called a 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. 3.

Applications

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

References

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

• Irving Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953) 839–853

Contemporary references are

Revised on July 30, 2014 23:50:32 by Rasmus Bentmann (134.76.82.183)