algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hilbert module is an abbreaviation both for a Hilbert C*-module (this entry) and the analogous notion of a Hilbert Q-module (see there), where is a quantale (or a locale, in particular).
The notion of Hilbert C*-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 C*-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)
such that
behaves like a positive definite Hermitian 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
.
In addition to the explicit -linearity in the second argument under right multiplicatojn
the axioms imply conjugate -linearity in the first argument and under left multiplication
Because:
First of all we have:
An ordinary complex Hilbert space is a Hilbert -module.
The archetypical class of examples of Hilbert C*-modules for commutative C*-algebras is the following. The general definition may be understood as the generalization of the structure of this example to non-commutative C*-algebras. See also remark below.
Let be a locally compact topological space and write for its C*-algebra of continuous functions vanishing at infinity.
Let be a fiber bundle of Hilbert spaces over , hence a canonically associated bundle to a unitary group-principal bundle. Then the space of continuous compactly supported sections is a Hilbert C*-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 .
Every C*-algebra is a Hilbert -module over itself when equipped by with the -valued inner product given simply by
In view of the archetypical example , example may be interpreted as exhibiting the trivial complex line bundle over whatever space is the C*-algebra of functions on (an actual topological space if is a commutative C*-algebra or else the noncommutative topology defined as the formal dual of ).
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.
In view of example we may think of example 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 for the trivial -bundle over is equivalent to . Example generalizes this to arbitrary C*-algebras .
For C*Alg and a Hilbert -module, def. , 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. , form a Banach star-algebra.
For itself regarded as a Hilbert -module as in example , this is the multiplier algebra of .
For two Hilbert C*-modules, an adjointable operator , def. , 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
and
are compact operators according to def. .
Hilbert C*-modules were introduced by Irving Kaplansky in
Contemporary references are
Last revised on September 12, 2023 at 07:53:50. See the history of this page for a list of all contributions to it.