Hilbert C*-module
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.
Definitions
[edit]Inner-product C*-modules
[edit]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).
- For all , in , and in :
- For all , in :
- from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form).
- For all in :
- in the sense of being a positive element of A, and
- (An element of a C*-algebra is said to be positive if it is self-adjoint with non-negative spectrum.)[8][9]
Hilbert C*-modules
[edit]An analogue to the Cauchy–Schwarz inequality holds for an inner-product -module :[10]
for , in .
On the pre-Hilbert module , define a norm by
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.
Examples
[edit]Hilbert spaces
[edit]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.
Vector bundles
[edit]If is a locally compact Hausdorff space and a vector bundle over with projection a Hermitian metric , then the space of continuous sections of is a Hilbert -module. Given sections of and the right action is defined by
and the inner product is given by
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*-algebras
[edit]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 .
The (algebraic) direct sum of copies of
can be made into a Hilbert -module by defining
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 module
[edit]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]
See also
[edit]Notes
[edit]- ^ 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. 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.
- ^ 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. doi:10.24033/asens.1677.
- ^ 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.
References
[edit]- Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press.
External links
[edit]- Weisstein, Eric W. "Hilbert C*-Module". MathWorld.
- Hilbert C*-Modules Home Page, a literature list