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.

They 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").

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 that used Hilbert C*-modules to construct a theory of induced 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.

Let ${\displaystyle A}$ be a C*-algebra (not assumed to be commutative or unital), its involution denoted by ${\displaystyle {}^{*}}$. An inner-product ${\displaystyle A}$-module (or pre-Hilbert ${\displaystyle A}$-module) is a complex linear space ${\displaystyle E}$ equipped with a compatible right ${\displaystyle A}$-module structure, together with a map

that satisfies the following properties:

An analogue to the Cauchy–Schwarz inequality holds for an inner-product ${\displaystyle A}$-module ${\displaystyle E}$:

for ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$.