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 Aristide Rieffel, the latter in a paper which 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. Wikipedia [ - ]

Hilbert C*-modules are mathematical objects which generalise the notion of a Hilbert space, in... [ + ]