# 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 A-module) is a complex linear space E which is equipped with a compatible right A-module structure, together with a map

${\displaystyle \langle \cdot ,\cdot \rangle :E\times E\rightarrow A}$

which satisfies the following properties:

• For all x, y, z in E, and α, β in C:
${\displaystyle \langle x,\alpha y+\beta z\rangle =\alpha \langle x,y\rangle +\beta \langle x,z\rangle }$
(i.e. the inner product is linear in its second argument).
• For all x, y in E, and a in A:
${\displaystyle \langle x,ya\rangle =\langle x,y\rangle a}$
• For all x, y in E:
${\displaystyle \langle x,y\rangle =\langle y,x\rangle ^{*},}$
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:
${\displaystyle \langle x,x\rangle \geq 0}$
and
${\displaystyle \langle x,x\rangle =0\iff x=0.}$
(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]

${\displaystyle \langle x,y\rangle \langle y,x\rangle \leq \Vert \langle x,x\rangle \Vert \langle y,y\rangle }$

for x, y in E.

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

${\displaystyle \Vert x\Vert =\Vert \langle x,x\rangle \Vert ^{\frac {1}{2}}.}$

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

${\displaystyle a_{\lambda }\rightarrow a\Rightarrow xa_{\lambda }\rightarrow xa.}$

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

${\displaystyle xe_{\lambda }\rightarrow x}$

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

Let

${\displaystyle \langle E,E\rangle =\operatorname {span} \{\langle x,y\rangle |x,y\in E\},}$

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.

## 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

${\displaystyle \langle f,h\rangle (x):=g(f(x),h(x)).}$

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

${\displaystyle A^{n}=\oplus _{1}^{n}A}$

can be made into a Hilbert A-module by defining

${\displaystyle \langle (a_{i}),(b_{i})\rangle =\sum a_{i}^{*}b_{i}.}$

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

${\displaystyle {\mathcal {H}}_{A}=\{(a_{i})|\sum a_{i}^{*}a_{i}{\text{ converges in }}A\}.}$

Given an inner product analogous to that on An, the resulting Hilbert A-module is called the standard Hilbert module.

## Notes

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9. 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.
10. 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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