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. Hilbert C*-modules 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.
Let be a C*-algebra (not assumed to be commutative or unital), its involution denoted by . An inner-product -module (or pre-Hilbert -module) is a complex linear space equipped with a compatible right -module structure, together with a map
that satisfies the following properties:
An analogue to the Cauchy–Schwarz inequality holds for an inner-product -module : [10]
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 (a net of self-adjoint elements of for which and tend to for each in ), 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.
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.
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 . [ citation needed]
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 (algebraic) 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.
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 ), the resulting Hilbert -module is called the standard Hilbert module over .
The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert -module there is an isometric isomorphism [11]
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. Hilbert C*-modules 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.
Let be a C*-algebra (not assumed to be commutative or unital), its involution denoted by . An inner-product -module (or pre-Hilbert -module) is a complex linear space equipped with a compatible right -module structure, together with a map
that satisfies the following properties:
An analogue to the Cauchy–Schwarz inequality holds for an inner-product -module : [10]
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 (a net of self-adjoint elements of for which and tend to for each in ), 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.
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.
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 . [ citation needed]
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 (algebraic) 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.
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 ), the resulting Hilbert -module is called the standard Hilbert module over .
The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert -module there is an isometric isomorphism [11]