# Hilbert C*-module

**Hilbert C*-modules** are mathematical objects which generalise the notion of a Hilbert space (which itself is a generalisation of Euclidean space), in that they endow a linear space with an "inner product" which 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").^{[1]} In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke^{[2]} and Marc Rieffel, the latter in a paper which used Hilbert C*-modules to construct a theory of induced representations of C*-algebras.^{[3]} Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory,^{[4]} and provide the right framework to extend the notion of Morita equivalence to C*-algebras.^{[5]} 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,^{[6]}^{[7]} and groupoid C*-algebras.

## Definitions

### 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**) is a complex linear space

*A*-module*E*which is equipped with a compatible right

*A*-module structure, together with a map

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.)^{[8]}^{[9]}

### Hilbert *A*-modules

An analogue to the Cauchy-Schwarz inequality holds for an inner-product *A*-module *E*:^{[10]}

for *x*, *y* in *E*.

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

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**. The Cauchy-Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

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

Similarly, if {*e _{λ}*} is an approximate unit for

*A*(a net of self-adjoint elements of

*A*for which

*ae*

_{λ}and

*e*

_{λ}

*a*tend to

*a*for each

*a*in

*A*), then for

*x*in

*E*

whence it follows that *EA* is dense in *E*, and *x*1 = *x* when *A* is unital.

Let

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.

## Examples

### 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*

can be made into a Hilbert *A*-module by defining

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

Given an inner product analogous to that on *A ^{n}*, the resulting Hilbert

*A*-module is called the

**standard Hilbert module**.

## See also

## Notes

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- ↑ In the case when
*A*is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to*A*. - ↑ This result in fact holds for semi-inner-product
*A*-modules, which may have non-zero elements*x*such that <*x*,*x*> = 0, as the proof does not rely on the nondegeneracy property.

## References

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## External links

- Weisstein, Eric W., "Hilbert C*-Module",
*MathWorld*. - Hilbert C*-Modules Home Page, a literature list