Hilbert C* module

Inner-product A-modules

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

⟨ ⋅ , ⋅ ⟩ : E × E → A

which satisfies the following properties:

For all x, y, z in E, and α, β in C:

(i.e. the inner product is linear in its second argument).

For all x, y in E, and a in A:

For all x, y in E:

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

For all x in E:

and (An element of a C*-algebra A is said to be positive if it is self-adjoint with non-negative spectrum.)

Hilbert A-modules

An analogue to the Cauchy-Schwarz inequality holds for an inner-product A-module E:

⟨ x , y ⟩ ⟨ y , x ⟩ ≤ ∥ ⟨ x , x ⟩ ∥ ⟨ y , y ⟩

for x, y in E.

On the pre-Hilbert module E, define a norm by

∥ x ∥ = ∥ ⟨ x , x ⟩ ∥ 1 2 .

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

The action of A on E is continuous: for all x in E

a λ → a ⇒ x a λ → x a .

Similarly, if {e_λ} is an approximate unit for A (a net of self-adjoint elements of A for which ae_λ and e_λa tend to a for each a in A), then for x in E

x e λ → x

whence it follows that EA is dense in E, and x1 = x when A is unital.

Let

⟨ E , E ⟩ = span ⁡ { ⟨ x , y ⟩ | x , y ∈ E } ,

then the closure of <E,E> is a two-sided ideal in A. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that E<E,E> is dense in E. In the case when <E,E> is dense in A, E is said to be full. This does not generally hold.

Hilbert spaces

A complex Hilbert space H is a Hilbert C-module under its inner product, the complex numbers being a C*-algebra with an involution given by complex conjugation.

Vector bundles

If X is a locally compact Hausdorff space and E a vector bundle over X with a Riemannian metric g, then the space of continuous sections of E is a Hilbert C(X)-module. The inner product is given by

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

C*-algebras

Any C*-algebra A is a Hilbert A-module under the inner product <a,b> = a*b. By the C*-identity, the Hilbert module norm coincides with C*-norm on A.

The (algebraic) direct sum of n copies of A

A n = ⊕ 1 n A

can be made into a Hilbert A-module by defining

⟨ ( a i ) , ( b i ) ⟩ = ∑ a i ∗ b i .

One may also consider the following subspace of elements in the countable direct product of A

H A = { ( a i ) | ∑ a i ∗ a i  converges in  A } .

Endowed with the obvious inner product (analogous to that of Aⁿ), the resulting Hilbert A-module is called the standard Hilbert module.