nLab
Hilbert module

Context

Functional analysis

Operator algebra

Index theory

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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

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