Hilbert C*-module

Hilbert C*-modules are <a href="/facts/Mathematical_object/Juhyow9M">mathematical objects</a> that generalise the notion of <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>
(which are themselves generalisations of <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>), 
in that they endow a <a href="/facts/Vector_space/M1pxMLx2">linear space</a> with an "<a href="/facts/Inner_product/HoyjElEy">inner product</a>" that takes values in a
<a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a>. 
They were first introduced in the work of <a href="/facts/Irving_Kaplansky/nRLyp7K9">Irving Kaplansky</a> in <a href="/facts/1953_in_science/HTuGFJUm">1953</a>, 
which developed the theory for <a href="/facts/Commutative/WaYxJLxd">commutative</a>, 
<a href="/facts/Unital_algebra/8T8ulWJb">unital algebras</a> 
(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 <a href="/facts/Marc_Rieffel/47FNj3vf">Marc Rieffel</a>, 
the latter in a paper that used Hilbert C*-modules to construct a theory of <a href="/facts/Induced_representation/Jh25PGDP">induced representations</a> of C*-algebras.
Hilbert C*-modules are crucial to Kasparov's formulation of <a href="/facts/KK-theory/ux2G1K9r">KK-theory</a>,
and provide the right framework to extend the notion
of <a href="/facts/Morita_equivalence/vEVRJSJw">Morita equivalence</a> to C*-algebras. 
They can be viewed as the generalization
of <a href="/facts/Vector_bundles/cXACAa1I">vector bundles</a> to noncommutative C*-algebras and as such play an important role in <a href="/facts/Noncommutative_geometry/WSwggL1p">noncommutative geometry</a>, 
notably in C*-algebraic quantum group theory, 
and <a href="/facts/Groupoid/hvj8wvWe">groupoid</a> C*-algebras.

Hilbert C*-module open-in-new

Hilbert C*-module