Hilbert C*-module

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"). [] it follows that

:$Vert\; x\; Vert\; =\; Vert\; langle\; x,\; x\; angle\; Vert^frac\{1\}\{2\}$

defines a norm on "E". When "E" is complete with respect to the metric induced by this norm, it is a Hilbert "A"-module or a Hilbert C*-module over the C*-algebra "A".

Examples

C*-algebras

Any C*-algebra "A" is a Hilbert "A"-module under the inner product <"a","b"> = "a"*"b". Here, the norm on the module coincides with the original norm on "A".

Hilbert spaces

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.

Vector bundles

If "X" is a locally compact Hausdorff space and "E" a vector bundle over "X" with a Riemannian metric "g", then the space of continuous sections of "E" is a Hilbert "C(X)"-module. The inner product is given by::$langle\; f,h\; angle\; (x):=g(f(x),h(x)).$

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".

Properties

*Let {"e"_λ}_{λ ∈ I} be an approximate unit for "A" (a net of self-adjoint elements of "A" for which "xe"_λ or "e"_λ"x" tend to "x" for each "x" in "A"), and let "E" be a Hilbert "A"-module. Then for "x" in "E"

::

:whence it follows that "EA" is dense in "E", and "x"1 = "x" when "A" is unital.

*Writing

::$langle\; E,\; E\; angle\; =\; operatorname\{span\}\; \{\; langle\; x,\; y\; angle\; |\; x,\; y\; in\; E\; \},$

:then the closure of <"E","E"> is a two-sided ideal in "A". Since this will also be a C*-algebra (and will therefore have an approximate unit), a calculation similar to the preceding one verifies that "E"<"E","E"> is dense in "E". In the case when <"E","E"> is dense in "E", "E" is said to be full. This does not generally hold.

See also

* C*-algebra
* Hilbert space
* KK-theory
* Operator algebra

Notes and references

*

External links

*
* [http://www.imn.htwk-leipzig.de/~mfrank/hilmod.html Hilbert C*-Modules Home Page] , a literature list