Direct Integral - Direct Integrals of Hilbert Spaces

Direct Integrals of Hilbert Spaces

The simplest example of a direct integral are the L2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions

Terminological note: The terminology adopted by the literature on the subject is followed here, according to which a measurable space X is referred to as a Borel space and the elements of the distinguished σ-algebra of X as Borel sets, regardless of whether or not the underlying σ-algebra comes from a topological space (in most examples it does). A Borel space is standard if and only if it is isomorphic to the underlying Borel space of a Polish space. Given a countably additive measure μ on X, a measurable set is one that differs from a Borel set by a null set. The measure μ on X is a standard measure if and only if there is a null set E such that its complement X − E is a standard Borel space. All measures considered here are σ-finite.

Definition. Let X be a Borel space equipped with a countably additive measure μ. A measurable family of Hilbert spaces on (X, μ) is a family {H_x}_{x∈ X}, which is locally equivalent to a trivial family in the following sense: There is a countable partition

by measurable subsets of X such that

where H_n is the canonical n-dimensional Hilbert space, that is

A cross-section of {H_x}_{x∈ X} is a family {s_x}_{x ∈ X} such that s_x ∈ H_x for all x ∈ X. A cross-section is measurable if and only if its restriction to each partition element X_n is measurable. We will identify measurable cross-sections s, t that are equal almost everywhere. Given a measurable family of Hilbert spaces

consists of equivalence classes (with respect to almost everywhere equality) of measurable square integrable cross-sections of {H_x}_{x∈ X}. This is a Hilbert space under the inner product

Given the local nature of our definition, many definitions applicable to single Hilbert spaces apply to measurable families of Hilbert spaces as well.

Remark. This definition is apparently more restrictive than the one given by von Neumann and discussed in Dixmier's classic treatise on von Neumann algebras. In this definition the Hilbert space fibers H_x are allowed to vary from point to point without having a local triviality requirement (local in a measure-theoretic sense). One of the main theorems of the von Neumann theory is to show that in fact the more general definition can be reduced to the simpler one given here.

Note that the direct integral of a measurable family of Hilbert spaces depends only on the measure class of the measure μ; more precisely:

Theorem. Suppose μ, ν are σ-finite countably additive measures on X that have the same sets of measure 0. Then the mapping

is a unitary operator