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

Let A be a C*-algebra (not assumed to be commutative or unital), its involution denoted by {}^*. An inner-product A-module (or pre-Hilbert A-module) is a complex linear space E equipped with a compatible right A-module structure, together with a map that satisfies the following properties:

Hilbert C*-modules

An analogue to the Cauchy–Schwarz inequality holds for an inner-product A-module E: for x, y in E. On the pre-Hilbert module E, define a norm by The norm-completion of E, still denoted by E, is said to be a Hilbert A-module or a Hilbert C-module over the C-algebra A**. The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion. The action of A on E is continuous: for all x in E Similarly, if (e_\lambda) is an approximate unit for A (a net of self-adjoint elements of A for which a e_\lambda and e_\lambda a tend to a for each a in A), then for x in E Whence it follows that EA is dense in E, and x 1_A = x when A is unital. Let then the closure of is a two-sided ideal in A. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that is dense in E. In the case when is dense in A, E is said to be full. This does not generally hold.

Examples

Hilbert spaces

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

Vector bundles

If X is a locally compact Hausdorff space and E a vector bundle over X with projection a Hermitian metric g, then the space of continuous sections of E is a Hilbert C(X)-module. Given sections of E and f \in C(X) 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 A = C(X) is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over X.

C*-algebras

Any C*-algebra A is a Hilbert A-module with the action given by right multiplication in A and the inner product. By the C*-identity, the Hilbert module norm coincides with C*-norm on A. The (algebraic) direct sum of n copies of A can be made into a Hilbert A-module by defining If p is a projection in the C*-algebra M_n(A), then pA^n is also a Hilbert A-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 A Endowed with the obvious inner product (analogous to that of A^n), the resulting Hilbert A-module is called the standard Hilbert module over A. The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert A-module E there is an isometric isomorphism

Maps between Hilbert modules

Let E and F be two Hilbert modules over the same C*-algebra A. 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 E and F. In the special case where A is \mathbb{C} these reduce to bounded and compact operators on Hilbert spaces respectively.

Adjointable maps

A map (not necessarily linear) is defined to be adjointable if there is another map, known as the adjoint of T, such that for every e \in E and f \in F, Both T and T^* are then automatically linear and also A-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, T^* is unique (if it exists) and itself adjointable with adjoint T. If is a second adjointable map, ST is adjointable with adjoint S^* T^. The adjointable operators E \to F form a subspace of, which is complete in the operator norm. In the case F = E, the space of adjointable operators from E to itself is denoted, and is a C-algebra.

Compact adjointable maps

Given e \in E and f \in F, 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 (closed, two-sided) ideal of.

C*-correspondences

If A and B are C*-algebras, an (A,B) C*-correspondence is a Hilbert B-module equipped with a left action of A 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

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

Toeplitz algebra of a correspondence

Given a C*-algebra A, and an (A,A) correspondence E, its Toeplitz algebra 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, crossed products by \mathbb{Z} , and the Cuntz algebras are all quotients of specific Toeplitz algebras.

Toeplitz representations

A Toeplitz representation of E in a C*-algebra D is a pair (S,\phi) of a linear map and a homomorphism such that

Toeplitz algebra

The Toeplitz algebra is the universal Toeplitz representation. That is, there is a Toeplitz representation (T, \iota) of E in such that if (S,\phi) is any Toeplitz representation of E (in an arbitrary algebra D) there is a unique *-homomorphism such that and.

Examples

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