Topics in Functional Analysis
AQFT and operator algebra
States and observables
The notion of Hilbert -module (or simply Hilbert module) is a generalization of the notion of Hilbert space where the algebra of complex numbers is replaced by a possibly more general C*-algebra . In particular a Hilbert -module has an inner product which takes values not in , but in , and such that complex conjugation is replaced by the star-operation in .
Hilbert -modules naturally appear as modules over groupoid convolution algebras. Refined to Hilbert C*-bimodules they serve as generalized homomorphism between C*-algebras in noncommutative topology, and, when further equipped with a left weak Fredholm module as cocycles in KK-theory.
For C*Alg, a Hilbert C*-module over is
a complex vector space ;
equipped with an action of from the right;
equipped with a sesquilinear map (linear in the second argument)
(the -valued inner product)
behaves like a positive definite inner product over in that for all and we have
(in the sense of positive elements in )
precisely if ;
is complete with respect to the norm
First of all we have:
An ordinary complex Hilbert space is a Hilbert -module.
The archetypical class of examples of Hilbert -modules for commutative C*-algebras is the following. The general definition 1 may be understood as the generalization of the structure of this example to non-cmmutative C*-algebras. See also remark 3 below.
Let be a locally compact topological space and write for its C*-algebra of continuous functions of compact support.
Let be a fiber bundle of Hilbert spaces over , hence an canonically associated bundle to a unitary group-principal bundle. Then the space of continuous compactly supported sections is a Hilbert -module over with -valued inner product the pointwise inner product in the Hilbert space fiber of :
Every Hilbert -module arises, up to isomorphism, as in example 2.
Every -algebra is a Hilbert -module over itself when equipped by with the -valued inner product given simply by
For C*Alg, let be the space of those sequences of elements in such that the series converges. This is a Hilbert -module when equipped with the degreewise -C*-representation, with the -valued inner product
and after completion with under the induced norm.
This is sometimes called the standard Hilbert -module over .
In view of example 2 we may think of example 4 as exhibiting the trivial countably-infinite dimensional Hilbert space bundle over the space dual to .
This is because the unitary group of an infinite-dimensional separable Hilbert space is contractible (by Kuiper's theorem), hence so is the classifying space, and so unitary -fiber bundles (over actual topological spaces) all trivializable. Since moreover the Hilbert module of example 2 for the trivial -bundle over is equivalent to . Example 4 generalizes this to arbitrary -algebras .
-algebras of adjointable operators on a Hilbert module
For C*Alg and a Hilbert -module, def. 1, a -linear operator is called adjointable if there is an adjoint operator? with respect to the -valued inner product in the sense that
The adjointable operators on a Hilbert -module, def. 2, form a Banach star-algebra.
For itself regarded as a Hilbert -module as in example 3, this is the multiplier algebra of .
Compact operators on a Hilbert -module
For two Hilbert -modules, an adjointable operator , def. 2, is of finite rank if it is of the form
for vectors and . is called a generalized compact operator if it is in the norm-closure of finite-rank operators.
Typically one writes for the space of generalized complact operators.
An operator is called a generalized Fredholm operator if there exists an operator (then called a parametrix for ) such that both
are compact operators according to def. 3.
- Kasparov’s KK-theory is formulated in terms of Hilbert (bi)modules
Section 13 of