nLab
Hilbert module

Context

Functional analysis

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Index theory

Hilbert modules

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

Idea

The notion of Hilbert C *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 AA. In particular a Hilbert AA-module has an inner product which takes values not in \mathbb{C}, but in AA, and such that complex conjugation is replaced by the star-operation in AA.

Hilbert C *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 BB \in C*Alg, a Hilbert C*-module over BB is

  1. a complex vector space HH;

  2. equipped with an action of BB from the right;

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

    ,:H×HB \langle -,-\rangle \colon H \times H \to B

    (the BB-valued inner product)

such that

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

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

    2. x,x0\langle x,x\rangle \geq 0 (in the sense of positive elements in BB)

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

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

  2. HH is complete with respect to the norm

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

Remark

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

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

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

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

Because:

vb,w =w,vb * =(w,vb) * =b *w,v * =b *v,w. \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 *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 XX be a locally compact topological space and write C 0(X)C_0(X) for its C*-algebra of continuous functions vanishing at infinity.

Let EXE \to X be a fiber bundle of Hilbert spaces over XX, hence a canonically associated bundle to a unitary group-principal bundle. Then the space Γ 0(E)\Gamma_0(E) of continuous compactly supported sections is a Hilbert C *C^\ast-module over C 0(X)C_0(X) with C 0(X)C_0(X)-valued inner product ,\langle -,-\rangle the pointwise inner product in the Hilbert space fiber of EE:

σ 1,σ 2(x)σ 1(y),σ 2(y) E yC 0(X),σ 1,σ 2Γ(E),xX. \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)C_0(X)-module arises, up to isomorphism, as in example 2.

Example

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

a 1,a 2a 1 *aA \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 AA is the C *C^\ast-algebra of functions on (an actual topological space if AA is a commutative C*-algebra or else the noncommutative topology defined as the formal dual of AA).

Example

For AA \in C*Alg, let 2A\ell^2 A be the space of those sequences {a nA} n\{a_n \in A\}_{n \in \mathbb{N}} of elements in AA such that the series na n *a n\sum_n a_n^\ast a_n converges. This is a Hilbert AA-module when equipped with the degreewise AA-C*-representation, with the AA-valued inner product

{a n},{b n} na 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 2A\ell^2 A is sometimes called the standard Hilbert AA-module over AA.

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 AA.

This is because the unitary group U()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 2()\mathcal{H} \simeq \ell^2(\mathbb{C}) the Hilbert module of example 2 for the trivial \mathcal{H}-bundle over C 0(X)C_0(X) is equivalent to 2(C 0(X))\ell^2(C_0(X)). Example 4 generalizes this to arbitrary C*C*-algebras AA.

Properties

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

Definition

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

F,=,F *. \langle F -, -\rangle = \langle -,F^\ast -\rangle \,.
Proposition

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

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

Compact operators on a Hilbert C *C^\ast-module

Definition

For H 1,H 2H_1, H_2 two Hilbert C *C^\ast-modules, an adjointable operator T:H 1H 2T \colon H_1 \to H_2, def. 2, is of finite rank if it is of the form

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

for vectors v iH 1v_i \in H_1 and w iH 2w_i \in H_2. TT 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)\mathcal{K}(H_1, H_2) for the space of generalized complact operators.

Fredholm operators

Definition

An operator F:H 1H 2F \colon H_1 \to H_2 is called a generalized Fredholm operator if there exists an operator S:H 2H 1S \colon H_2 \to H_1 (then called a parametrix for FF) such that both

FSid H 2 F \circ S - id_{H_2} and SFid H 1S \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 *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

Last revised on September 10, 2014 at 14:41:58. See the history of this page for a list of all contributions to it.