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").
[1] In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke[2] and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras.
[3] Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory,[4] and provide the right framework to extend the notion of Morita equivalence to C*-algebras.
[5] 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,[6][7] and groupoid C*-algebras.
be a C*-algebra (not assumed to be commutative or unital), its involution denoted by
-module structure, together with a map that satisfies the following properties: An analogue to the Cauchy–Schwarz inequality holds for an inner-product
The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.
Two-sided ideals are C*-subalgebras and therefore possess approximate units.
-module under scalar multipliation by complex numbers and its inner product.
is a locally compact Hausdorff space and
the right action is defined by and the inner product is given by The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra
By the C*-identity, the Hilbert module norm coincides with C*-norm on
One may also consider the following subspace of elements in the countable direct product of
Endowed with the obvious inner product (analogous to that of
-module is called the standard Hilbert module over
The fact that there is a unique separable Hilbert space has a generalization to Hilbert modules in the form of the Kasparov stabilization theorem, which states that if
The adjointable and compact adjointable operators are subspaces of this Banach space defined using the inner product structures on
these reduce to bounded and compact operators on Hilbert spaces respectively.
The closed graph theorem can be used to show that they are also bounded.
Analogously to the adjoint of operators on Hilbert spaces,
(NB: Some authors require the left action to be non-degenerate instead.)
These objects are used in the formulation of Morita equivalence for C*-algebras, see applications in the construction of Toeplitz and Cuntz-Pimsner algebras,[14] and can be employed to put the structure of a bicategory on the collection of C*-algebras.
correspondence, the algebraic tensor product
as vector spaces inherits left and right
-valued sesquilinear form defined on pure tensors by This is positive semidefinite, and the Hausdorff completion of
[16] The collection of C*-algebras can then be endowed with the structure of a bicategory, with C*-algebras as objects,
, and isomorphisms of correspondences (bijective module maps that preserve inner products) as 2-arrows.
[18] In particular, graph algebras , crossed products by
isometries with mutually orthogonal range projections.