Physics:Hilbert C*-module

Short description: Mathematical objects that generalise the notion of Hilbert spaces

Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces (which are themselves generalisations of Euclidean space), in that they endow a linear space with an "inner product" that takes values in a C*-algebra. Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital"). In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.

Definitions

Inner-product C*-modules

Let [math]\displaystyle{ A }[/math] be a C*-algebra (not assumed to be commutative or unital), its involution denoted by [math]\displaystyle{ {}^* }[/math]. An inner-product [math]\displaystyle{ A }[/math]-module (or pre-Hilbert [math]\displaystyle{ A }[/math]-module) is a complex linear space [math]\displaystyle{ E }[/math] equipped with a compatible right [math]\displaystyle{ A }[/math]-module structure, together with a map

[math]\displaystyle{ \langle \, \cdot \, , \, \cdot \,\rangle_A : E \times E \rightarrow A }[/math]

that satisfies the following properties:

• For all [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], [math]\displaystyle{ z }[/math] in [math]\displaystyle{ E }[/math], and [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \beta }[/math] in [math]\displaystyle{ \mathbb{C} }[/math]:

[math]\displaystyle{ \langle x ,y \alpha + z \beta \rangle_A = \langle x, y \rangle_A \alpha + \langle x, z \rangle_A \beta }[/math]

(i.e. the inner product is [math]\displaystyle{ \mathbb{C} }[/math]-linear in its second argument).

• For all [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math] in [math]\displaystyle{ E }[/math], and [math]\displaystyle{ a }[/math] in [math]\displaystyle{ A }[/math]:

[math]\displaystyle{ \langle x, y a \rangle_A = \langle x, y \rangle_A a }[/math]

• For all [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math] in [math]\displaystyle{ E }[/math]:

[math]\displaystyle{ \langle x, y \rangle_A = \langle y, x \rangle_A^*, }[/math]

from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form).

• For all [math]\displaystyle{ x }[/math] in [math]\displaystyle{ E }[/math]:

[math]\displaystyle{ \langle x, x \rangle_A \geq 0 }[/math]

in the sense of being a positive element of A, and

[math]\displaystyle{ \langle x, x \rangle_A = 0 \iff x = 0. }[/math]

(An element of a C*-algebra [math]\displaystyle{ A }[/math] is said to be positive if it is self-adjoint with non-negative spectrum.)

Hilbert C*-modules

An analogue to the Cauchy–Schwarz inequality holds for an inner-product [math]\displaystyle{ A }[/math]-module [math]\displaystyle{ E }[/math]:

[math]\displaystyle{ \langle x, y \rangle_A \langle y, x \rangle_A \leq \Vert \langle y, y \rangle_A \Vert \langle x, x \rangle_A }[/math]

for [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math] in [math]\displaystyle{ E }[/math].

On the pre-Hilbert module [math]\displaystyle{ E }[/math], define a norm by

[math]\displaystyle{ \Vert x \Vert = \Vert \langle x, x \rangle_A \Vert^\frac{1}{2}. }[/math]

The norm-completion of [math]\displaystyle{ E }[/math], still denoted by [math]\displaystyle{ E }[/math], is said to be a Hilbert [math]\displaystyle{ A }[/math]-module or a Hilbert C*-module over the C*-algebra [math]\displaystyle{ A }[/math]. The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

The action of [math]\displaystyle{ A }[/math] on [math]\displaystyle{ E }[/math] is continuous: for all [math]\displaystyle{ x }[/math] in [math]\displaystyle{ E }[/math]

[math]\displaystyle{ a_{\lambda} \rightarrow a \Rightarrow xa_{\lambda} \rightarrow xa. }[/math]

Similarly, if [math]\displaystyle{ (e_\lambda) }[/math] is an approximate unit for [math]\displaystyle{ A }[/math] (a net of self-adjoint elements of [math]\displaystyle{ A }[/math] for which [math]\displaystyle{ a e_\lambda }[/math] and [math]\displaystyle{ e_\lambda a }[/math] tend to [math]\displaystyle{ a }[/math] for each [math]\displaystyle{ a }[/math] in [math]\displaystyle{ A }[/math]), then for [math]\displaystyle{ x }[/math] in [math]\displaystyle{ E }[/math]

[math]\displaystyle{ xe_\lambda \rightarrow x. }[/math]

Whence it follows that [math]\displaystyle{ EA }[/math] is dense in [math]\displaystyle{ E }[/math], and [math]\displaystyle{ x 1_A = x }[/math] when [math]\displaystyle{ A }[/math] is unital.

Let

[math]\displaystyle{ \langle E, E \rangle_A = \operatorname{span} \{ \langle x, y \rangle_A \mid x, y \in E \}, }[/math]

then the closure of [math]\displaystyle{ \langle E, E \rangle_A }[/math] is a two-sided ideal in [math]\displaystyle{ A }[/math]. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that [math]\displaystyle{ E \langle E, E \rangle_A }[/math] is dense in [math]\displaystyle{ E }[/math]. In the case when [math]\displaystyle{ \langle E , E \rangle_A }[/math] is dense in [math]\displaystyle{ A }[/math], [math]\displaystyle{ E }[/math] is said to be full. This does not generally hold.

Examples

Hilbert spaces

Since the complex numbers [math]\displaystyle{ \mathbb{C} }[/math] are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space [math]\displaystyle{ \mathcal{H} }[/math] is a Hilbert [math]\displaystyle{ \mathbb{C} }[/math]-module under scalar multipliation by complex numbers and its inner product.

Vector bundles

If [math]\displaystyle{ X }[/math] is a locally compact Hausdorff space and [math]\displaystyle{ E }[/math] a vector bundle over [math]\displaystyle{ X }[/math] with projection [math]\displaystyle{ \pi \colon E \to X }[/math] a Hermitian metric [math]\displaystyle{ g }[/math], then the space of continuous sections of [math]\displaystyle{ E }[/math] is a Hilbert [math]\displaystyle{ C(X) }[/math]-module. Given sections [math]\displaystyle{ \sigma, \rho }[/math] of [math]\displaystyle{ E }[/math] and [math]\displaystyle{ f \in C(X) }[/math] the right action is defined by

[math]\displaystyle{ \sigma f (x) = \sigma(x) f(\pi(x)), }[/math]

and the inner product is given by

[math]\displaystyle{ \langle \sigma,\rho\rangle_{C(X)} (x):=g(\sigma(x),\rho(x)). }[/math]

The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra [math]\displaystyle{ A = C(X) }[/math] is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over [math]\displaystyle{ X }[/math].

C*-algebras

Any C*-algebra [math]\displaystyle{ A }[/math] is a Hilbert [math]\displaystyle{ A }[/math]-module with the action given by right multiplication in [math]\displaystyle{ A }[/math] and the inner product [math]\displaystyle{ \langle a , b \rangle = a^*b }[/math]. By the C*-identity, the Hilbert module norm coincides with C*-norm on [math]\displaystyle{ A }[/math].

The (algebraic) direct sum of [math]\displaystyle{ n }[/math] copies of [math]\displaystyle{ A }[/math]

[math]\displaystyle{ A^n = \bigoplus_{i=1}^n A }[/math]

can be made into a Hilbert [math]\displaystyle{ A }[/math]-module by defining

[math]\displaystyle{ \langle (a_i), (b_i) \rangle_A = \sum_{i=1}^n a_i^* b_i. }[/math]

If [math]\displaystyle{ p }[/math] is a projection in the C*-algebra [math]\displaystyle{ M_n(A) }[/math], then [math]\displaystyle{ pA^n }[/math] is also a Hilbert [math]\displaystyle{ A }[/math]-module with the same inner product as the direct sum.

The standard Hilbert module

One may also consider the following subspace of elements in the countable direct product of [math]\displaystyle{ A }[/math]

[math]\displaystyle{ \ell_2(A)= \mathcal{H}_A = \Big\{ (a_i) | \sum_{i=1}^{\infty} a_i^{*}a_i\text{ converges in }A \Big\}. }[/math]

Endowed with the obvious inner product (analogous to that of [math]\displaystyle{ A^n }[/math]), the resulting Hilbert [math]\displaystyle{ A }[/math]-module is called the standard Hilbert module over [math]\displaystyle{ A }[/math].

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert [math]\displaystyle{ A }[/math]-module [math]\displaystyle{ E }[/math] there is an isometric isomorphism [math]\displaystyle{ E \oplus \ell^2(A) \cong \ell^2(A). }[/math]

See also

• Operator algebra

References

• Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press. 

External links

• Weisstein, Eric W.. "Hilbert C*-Module". http://mathworld.wolfram.com/HilbertC-Star-Module.html. 
• Hilbert C*-Modules Home Page, a literature list