Hilbert C*-module

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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 Aristide 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.

1 Definitions
- 1.1 Inner-product A-modules
- 1.2 Hilbert A-modules
2 Examples
3 Properties
4 See also
5 Notes and references
6 External links

[edit] Definitions

[edit] 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 a right A-module, together with a map

$\langle \cdot, \cdot \rangle : E \times E \rightarrow A$

which satisfies the following properties:

For all x, y, z in E, and α, β in C:

$\langle x, \alpha y + \beta z \rangle = \alpha \langle x, y \rangle + \beta \langle x, z \rangle$

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

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

$\langle x, ya \rangle = \langle x, y \rangle a$

For all x, y in E:

$\langle x, y \rangle = \langle y, x \rangle^*,$

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:

$\langle x, x \rangle \geq 0$

and

$\langle x, x \rangle = 0 \iff x = 0.$

(Here, the ordering on A is obtained by defining the "positive" elements to be all self-adjoint x of non-negative spectrum.)^[8]^[9]

[edit] Hilbert A-modules

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

$\langle x, y \rangle \langle y, x \rangle \leq \Vert \langle x, x \rangle \Vert \langle y, y \rangle$

for x, y in E. Since for all positive, self-adjoint elements x, y in A, it holds that ||x|| ≤ ||y|| if x ≤ y,^[11] it follows that

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

defines a norm on E. When E is complete with respect to the metric induced by this norm, it is a Hilbert A-module or a Hilbert C*-module over the C*-algebra A.

[edit] Examples

[edit] C*-algebras

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

[edit] Hilbert spaces

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

[edit] 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

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

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.

[edit] Properties

Let {e_λ}_{λ ∈ I} be an approximate unit for A (a net of self-adjoint elements of A for which xe_λ or e_λx tend to x for each x in A), and let E be a Hilbert A-module. Then for x in E

$\begin{align} \Vert x - xe_\lambda \Vert ^2 & = \langle x - xe_\lambda, x - xe_\lambda \rangle \\ & = \langle x, x \rangle - e_\lambda \langle x, x \rangle - \langle x, x \rangle e_\lambda + e_\lambda \langle x, x \rangle e_\lambda \ \xrightarrow{\lambda} \ 0, \\ \end{align}$

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

Writing

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

then the closure of <E,E> is a two-sided ideal in A. Since this will also be a C*-algebra (and will therefore have an approximate unit), a calculation similar to the preceding one verifies that E<E,E> is dense in E. In the case when <E,E> is dense in E, E is said to be full. This does not generally hold.

[edit] See also

[edit] Notes and references

^ Kaplansky, I. (1953). "Modules over operator algebras". American Journal of Mathematics 75 (4): 839–853. doi:10.2307/2372552.
^ Paschke, W. L. (1973). "Inner product modules over B*-algebras". Transactions of the American Mathematical Society 182: 443–468. doi:10.2307/1996542.
^ Rieffel, M. A. (1974). "Induced representations of C*-algebras". Advances in Mathematics 13: 176–257. Elsevier. doi:10.1016/0001-8708(74)90068-1.
^ Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory 4: 133–150. Theta Foundation.
^ Rieffel, M. A. (1982). "Morita equivalence for operator algebras". Proceedings of Symposia in Pure Mathematics 38: 176–257. American Mathematical Society.
^ Baaj, S.; Skandalis, G. (1993). "Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres". Annales Scientifiques de l'École Normale Supérieure 26 (4): 425–488.
^ Woronowicz, S. L. (1991). "Unbounded elements affiliated with C*-algebras and non-compact quantum groups". Communications in Mathematical Physics 136: 399–432. doi:10.1007/BF02100032.
^ Arveson, William (1976). An Invitation to C*-Algebras. Springer-Verlag, p. 35.
^ 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.
^ Arveson, William (1976). An Invitation to C*-Algebras. Springer-Verlag, p. 39.

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

[edit] External links

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

Categories: C*-algebras | Operator theory | Theoretical physics

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