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

that satisfies the following properties:

For all , ,  in , and ,  in :

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

For all ,  in , and  in :

For all ,  in :

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

For all  in :

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

(An element of a C*-algebra  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 -module :

for ,  in .

On the pre-Hilbert module , define a norm by

The norm-completion of , still denoted by , is said to be a Hilbert -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.

The action of  on  is continuous: for all  in 

Similarly, if  is an approximate unit for  (a net of self-adjoint elements of  for which  and  tend to  for each  in ), then for  in 

Whence it follows that  is dense in , and  when  is unital.
 
Let

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

Examples

Hilbert spaces 
Since the complex numbers  are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space  is a Hilbert -module under scalar multipliation by complex numbers and its inner product.

Vector bundles
If  is a locally compact Hausdorff space and  a vector bundle over  with projection  a Hermitian metric , then the space of continuous sections of  is a Hilbert -module. Given sections  of   and  the right action is defined by

and the inner product is given by

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

C*-algebras 
Any C*-algebra  is a Hilbert -module with the action given by right multiplication in  and the inner product .  By the C*-identity, the Hilbert module norm coincides with C*-norm on .

The (algebraic) direct sum of  copies of 

can be made into a Hilbert -module by defining

If  is a projection in the C*-algebra , then  is also a Hilbert -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 

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

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert -module  there is an isometric isomorphism

See also 
 Operator algebra

Notes

References

External links 
 
 Hilbert C*-Modules Home Page, a literature list

C*-algebras
Operator theory
Theoretical physics