Hilbert C*-modules are mathematical objects which generalise the notion of a Hilbert 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. In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper which used Hilbert C*-modules to construct a theory ofinduced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. 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, and groupoid C*-algebras.
Definitions
Inner-product ''A''-modules
Let A be a C*-algebra, its involution denoted by *. An inner-product A-module is a complex linear spaceE which is equipped with a compatible right A-module structure, together with a map which satisfies the following properties:
For all x, y, z in E, and α, β in C:
For all x, y in E, and a in A:
For all x, y in E:
For all x in E:
Hilbert ''A''-modules
An analogue to the Cauchy–Schwarz inequality holds for an inner-product A-module E: for x, y in E. On the pre-Hilbert module E, define a norm by The norm-completion of E, still denoted by E, is said to be a Hilbert A-module or a Hilbert C*-module over the C*-algebra A. The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion. The action of A on E is continuous: for all x in E Similarly, if is an approximate unit for A, then for x in E whence it follows that EA is dense in E, and x1 = x when A is unital. Let then the closure of <E,E> is a two-sided ideal in A. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that E<E,E> is dense in E. In the case when <E,E> is dense in A, E is said to be full. This does not generally hold.
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.
Any C*-algebra A is a Hilbert A-module under the inner product <a,b> = a*b. By the C*-identity, the Hilbert module norm coincides with C*-norm on A. The direct sum of n copies of A can be made into a Hilbert A-module by defining One may also consider the following subspace of elements in the countable direct product of A Endowed with the obvious inner product, the resulting Hilbert A-module is called the standard Hilbert module.