Hilbert C*-module

Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces
,
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
.
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

Let be a C*-algebra, its involution denoted by. An inner-product -module is a complex linear space equipped with a compatible right -module structure, together with a map
that satisfies the following properties:

For all,, in, and, in :
For all, in, and in :
For all, in :
For all in :

Hilbert C*-modules

An analogue to the Cauchy–Schwarz inequality holds for an inner-product -module :
for, in.
On the pre-Hilbert module, define a norm by
The norm-completion of, still denoted by, is said to be a Hilbert -module or a Hilbert C*-module over the C*-algebra .
The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.
The action of on is continuous: for all in
Similarly, if is an approximate unit for, then for in
Whence it follows that is dense in, and when is unital.
Let
then the closure of is a two-sided ideal in. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that is dense in. In the case when is dense in, is said to be full. This does not generally hold.

Examples

Hilbert spaces

Since the complex numbers are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space is a Hilbert -module under scalar multipliation by complex numbers and its inner product.

Vector bundles

If is a locally compact Hausdorff space and a vector bundle over with projection a Hermitian metric, then the space of continuous sections of is a Hilbert -module. Given sections of 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 is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over.

C*-algebras

Any C*-algebra is a Hilbert -module with the action given by right multiplication in and the inner product. By the C*-identity, the Hilbert module norm coincides with C*-norm on.
The direct sum of copies of
can be made into a Hilbert -module by defining
If is a projection in the C*-algebra, then is also a Hilbert -module with the same inner product as the direct sum.

The standard Hilbert module

One may also consider the following subspace of elements in the countable direct product of
Endowed with the obvious inner product, the resulting Hilbert -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 is a countably generated Hilbert -module, there is an isometric isomorphism

Maps between Hilbert modules

Let and be two Hilbert modules over the same
C*-algebra. These are then Banach spaces, so it is possible to
speak of the Banach space of bounded linear maps,
normed by the operator norm.
The adjointable and compact adjointable operators are subspaces of this Banach space
defined using the inner product structures on and.
In the special case where is these reduce to
bounded and compact operators on Hilbert spaces respectively.

Adjointable maps

A map
is defined to be adjointable if there
is another map, known as the adjoint
of, such that for every
and,
Both and are then automatically linear
and also -module maps. The
closed graph theorem can be used to show that they are also bounded.
Analogously to the adjoint of operators on Hilbert spaces,
is unique and itself adjointable with adjoint. If
is a second adjointable map, is adjointable with adjoint
.
The adjointable operators form a subspace
of, which is complete in the operator norm.
In the case, the space of
adjointable operators from to itself is denoted, and is a
C*-algebra.

Compact adjointable maps

Given and, the map
is defined, analogously to the
rank one operators of Hilbert spaces, to be
This is adjointable with adjoint.
The compact adjointable operators are defined to be the closed span
of
in.
As with the bounded operators, is denoted
. This is a
ideal of

C*-correspondences

If and are C*-algebras, an C*-correspondence
is a Hilbert -module equipped with a left action of by
adjointable maps that is faithful. 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

If is an and a correspondence,
the algebraic tensor product of and
as vector spaces inherits left and right - and -module
structures respectively.
It can also be endowed with the -valued sesquilinear form defined on
pure tensors by
This is positive semidefinite, and the Hausdorff completion of
in the resulting seminorm is denoted. The left- and right-actions of
and extend to make this an correspondence.
The collection of C*-algebras can then be endowed with
the structure of a bicategory, with C*-algebras as
objects, correspondences as
arrows, and isomorphisms of correspondences as 2-arrows.

Toeplitz algebra of a correspondence

Given a C*-algebra, and an correspondence,
its Toeplitz algebra is defined as the universal algebra
for Toeplitz representations.
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 , crossed products by , and the
Cuntz algebras are all quotients of specific Toeplitz algebras.

Toeplitz representations

A Toeplitz representation of in a C*-algebra
is a pair
of a linear map and a homomorphism
such that

is "isometric":
resembles a bimodule map:

Toeplitz algebra

The Toeplitz algebra is the universal Toeplitz representation.
That is, there is a Toeplitz representation of
in such that if is any Toeplitz representation
of there is a unique *-homomorphism
such that
and.

Examples

If is taken to be the algebra of complex numbers, and
the vector space, endowed with the natural
-bimodule structure, the corresponding Toeplitz algebra
is the universal algebra generated by isometries with mutually orthogonal
range projections.
In particular, is the universal algebra generated by
a single isometry, which is the classical Toeplitz algebra.