Mathematical objects that generalise the notion of Hilbert spaces
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").[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 that 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.
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).
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.
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.
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 . [citation needed]
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 .
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 [11]
^Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. 4. Theta Foundation: 133–150.
^Rieffel, M. A. (1982). "Morita equivalence for operator algebras". Proceedings of Symposia in Pure Mathematics. 38. American Mathematical Society: 176–257.
^Arveson, William (1976). An Invitation to C*-Algebras. Springer-Verlag. p. 35.
^In the case when is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to .
^This result in fact holds for semi-inner-product -modules, which may have non-zero elements such that , as the proof does not rely on the nondegeneracy property.
^Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. 4. ThetaFoundation: 133–150.
Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press.