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Wikipedia / 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.

Definitions

Inner-product C*-modules

be a C*-algebra (not assumed to be commutative or unital), its involution. An -module (or -module) is a complex equipped with a compatible right A-module structure, together with a map

that satisfies the following properties:

  • :

  • (i.e.-linear in its second argument).

  • :

    :

  • from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form).

  • :

  • in the sense of being a positive element of A, and

    is said to be positive if it is self-adjoint with non-negative spectrum.)

Hilbert C*-modules

An analogue to the Cauchy–Schwarz inequality:

.

, define a norm by

, is said to be a -module or a . The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

is an approximate unit (a net

is dense is unital.

Let

then the closure is said to be full. This does not generally hold.

Examples

Hilbert spaces

are a C*-algebra with an involution given by complex conjugation-module under scalar multipliation by complex numbers and its inner product.

Vector bundles

is a locally compact Hausdorff space a vector bundle a Hermitian metric the right action is defined by

and the inner product is given by

.

C*-algebras

.

The (algebraic) direct sum

-module by defining

-module with the same inner product as the direct sum.

The standard Hilbert module

-module is called the .

The fact that there is a unique separable Hilbert space has a generalization to Hilbert modules in the form of the Kasparov stabilization theorem

Maps between Hilbert modules

. These are then Banach spaces, normed by the operator norm.

The adjointable.

these reduce to bounded and compact operators on Hilbert spaces respectively.

Adjointable maps

,

-module maps. The closed graph theorem can be used to show that they are also bounded.

.

, which is complete in the operator norm.

, and is a C*-algebra.

Compact adjointable maps

is defined, analogously to the rank one operators of Hilbert spaces, to be

.

are defined to be the closed span of

.

.

C*-correspondences

by adjointable maps that is faithful. (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, and can be employed to put the structure of a bicategory on the collection of C*-algebras.

Tensor products and the bicategory of correspondences

-module structures respectively.

-valued sesquilinear form defined on pure tensors by

correspondence.

, and isomorphisms of correspondences (bijective module maps that preserve inner products) as 2-arrows.

Toeplitz algebra of a correspondence

is defined as the universal algebra for Toeplitz representations (defined below).

The classical Toeplitz algebra can be recovered as a special case, and the Cuntz-Pimsner algebras are defined as particular quotients of Toeplitz algebras.

In particular, graph algebras, \mathbb{Z}, and the Cuntz algebras are all quotients of specific Toeplitz algebras.

Toeplitz representations

A Toeplitz representation of a linear map such that

  • is "isometric":

  • ,

  • resembles a bimodule map:

  • .

Toeplitz algebra

.

Examples

-bimodule structure, the corresponding Toeplitz algebra is the universal algebra isometries with mutually orthogonal range projections.

is the universal algebra generated by a single isometry, which is the classical Toeplitz algebra.

See also

Notes

References