Hilbert Module

Hilbert C*-modules are mathematical objects that generalise the notion of a Hilbert space (which itself is a generalisation 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"). 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 ''A''-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
:$\langle \cdot, \cdot \rangle : E \times E \rightarrow A$
that satisfies the following properties:

*For all ''x'', ''y'', ''z'' in ''E'', and ''α'', ''β'' in C:

::$\langle x, y \alpha + z \beta \rangle = \langle x, y \rangle \alpha + \langle x, z \rangle \beta$

:(''i.e.'' the inner product is linear in its second argument).

*For all ''x'', ''y'' in ''E'', and ''a'' in ''A'':
::$\langle x, y a \rangle = \langle x, y \rangle a$

*For all ''x'', ''y'' in ''E'':

::$\langle x, y \rangle = \langle y, x \rangle^*,$

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

*For all ''x'' in ''E'':

::$\langle x, x \rangle \geq 0$

:and

::$\langle x, x \rangle = 0 \iff x = 0.$

:(An element of a C*-algebra ''A'' is said to be ''positive'' if it is self-adjoint with non-negative spectrum.)

Hilbert ''A''-modules

An analogue to the Cauchy–Schwarz inequality holds for an inner-product ''A''-module ''E'':This result in fact holds for semi-inner-product ''A''-modules, which may have non-zero elements ''x'' such that <''x'',''x''> = 0, as the proof does not rely on the nondegeneracy property.

:$\langle x, y \rangle \langle y, x \rangle \leq \Vert \langle y, y \rangle \Vert \langle x, x \rangle$

for ''x'', ''y'' in ''E''.

On the pre-Hilbert module ''E'', define a norm by

:$\Vert x \Vert = \Vert \langle x, x \rangle \Vert^\frac.$

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''

:$a_ \rightarrow a \Rightarrow xa_ \rightarrow xa.$

Similarly, if is an approximate unit for ''A'' (a net of self-adjoint elements of ''A'' for which ''ae''_λ and ''e''_λ''a'' tend to ''a'' for each ''a'' in ''A''), then for ''x'' in ''E''

:$xe_\lambda \rightarrow x$

whence it follows that ''EA'' is dense in ''E'', and ''x''1 = ''x'' when ''A'' is unital.

Let

:$\langle E, E \rangle = \operatorname \,$

then the closure of <''E'',''E''> is a two-sided ideal in ''A''. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that ''E''<''E'',''E''> is dense in ''E''. In the case when <''E'',''E''> is dense in ''A'', ''E'' is said to be full. This does not generally hold.

Examples

Hilbert spaces

A complex Hilbert space ''H'' is a Hilbert C-module under its inner product, the complex numbers being a C*-algebra with an involution given by complex conjugation.

Vector bundles

If ''X'' is a locally compact Hausdorff space and ''E'' a vector bundle over ''X'' with a Riemannian metric ''g'', then the space of continuous sections of ''E'' is a Hilbert ''C(X)''-module. The inner product is given by
::$\langle f,h\rangle (x):=g(f(x),h(x)).$

The converse holds as well: Every countably generated Hilbert C*-module over a commutative 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 under the inner product <''a'',''b''> = ''a''*''b''. By the C*-identity, the Hilbert module norm coincides with C*-norm on ''A''.

The (algebraic) direct sum of ''n'' copies of ''A''

:$A^n = \oplus_1^n A$

can be made into a Hilbert ''A''-module by defining

:$\langle (a_i), (b_i) \rangle = \sum a_i^* b_i.$

One may also consider the following subspace of elements in the countable direct product of ''A''

:$\ell_2(A)= \mathcal_A = \.$

Endowed with the obvious inner product (analogous to that of ''Aⁿ''), the resulting Hilbert ''A''-module is called the standard Hilbert module.

See also

* Operator algebra

Notes

References

*

External links

*

Hilbert C*-Modules Home Page
a literature list

{{DEFAULTSORT:Hilbert C-module
C*-algebras
Operator theory
Theoretical physics