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.
Definitionsedit
Inner-product C*-modulesedit
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.
Examplesedit
Hilbert spacesedit
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]
C*-algebrasedit
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 .
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 moduleedit
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]
^Rieffel, M. A. (1974). "Induced representations of C*-algebras". Advances in Mathematics. 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. 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.
^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. S2CID 118184597.
^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.
Referencesedit
Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press.