Hilbert C*-module
Hilbert C*-modules are
Definitions
Inner-product C*-modules
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
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
Hilbert spaces
Since the complex numbers are a C*-algebra with an involution given by
Vector bundles
If is a
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
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
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
Notes
- JSTOR 2372552.
- JSTOR 1996542.
- .
- ^ 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.
- .
- 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 nondegeneracyproperty.
- ^ Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. 4. ThetaFoundation: 133–150.
References
- 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
- Weisstein, Eric W. "Hilbert C*-Module". MathWorld.
- Hilbert C*-Modules Home Page, a literature list