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"). 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 which 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.
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
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:
- (An element of a C*-algebra A is said to be positive if it is self-adjoint with non-negative spectrum.)
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 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
whence it follows that EA is dense in E, and x1 = x when A is unital.
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.
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.
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.
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 subspace of elements in the countable direct product of A
Endowed with the obvious inner product (analogous to that of An), the resulting Hilbert A-module is called the standard Hilbert module.
- Kaplansky, I. (1953). "Modules over operator algebras". American Journal of Mathematics. 75 (4): 839–853. doi:10.2307/2372552. JSTOR 2372552.
- Paschke, W. L. (1973). "Inner product modules over B*-algebras". Transactions of the American Mathematical Society. 182: 443–468. doi:10.2307/1996542. JSTOR 1996542.
- Rieffel, M. A. (1974). "Induced representations of C*-algebras". Advances in Mathematics. Elsevier. 13 (2): 176–257. doi:10.1016/0001-8708(74)90068-1.
- Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. Theta Foundation. 4: 133–150.
- Rieffel, M. A. (1982). "Morita equivalence for operator algebras". Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 38: 176–257.
- 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 (2): 399–432. Bibcode:1991CMaPh.136..399W. 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.
- Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press.