Measurable field of Hilbert spaces - Imported Knowlpedia Content

Definition

Let $(X,\Sigma)$ be a measurable space. A measurable field of Hilbert spaces consists of Hilbert spaces $H_x$ and a linear subspace $\mathcal M\subseteq\prod_{x\in X}H_x$ of sections such that:

$x\mapsto\lVert\xi(x)\rVert$ is measurable for every $\xi\in\mathcal M$ ;
a section $\eta$ lies in $\mathcal M$ whenever $x\mapsto\langle\eta(x),\xi(x)\rangle$ is measurable for every $\xi\in\mathcal M$ ; and
some sequence $(\xi_n)$ in $\mathcal M$ has $\{\xi_n(x):n\geq1\}$ dense in every $H_x$ .

The elements of $\mathcal M$ are the measurable sections.

Fundamental sequences and coordinates

A sequence as in the third axiom is called a fundamental sequence. Measurable fiberwise Gram–Schmidt operations turn one into sections that form an orthonormal basis after zero vectors are omitted. Relative to such sections, measurability can be checked through scalar coordinate functions. The countability requirement forces every fiber to be separable and is what makes direct-integral constructions manageable Takesaki, Chapter IV, §8.

Direct integrals

Given a measure space $(X,\Sigma,\mu)$ , one takes measurable sections satisfying

\int_X\lVert\xi(x)\rVert^2\,d\mu(x)<\infty

and identifies sections equal almost everywhere. The resulting Hilbert space, denoted $\int_X^\oplus H_x\,d\mu(x)$ , has inner product obtained by integrating the fiber inner products. Measurable fields therefore supply the varying-fiber data behind continuous decompositions of operators and representations.

Examples and conventions

A fixed separable Hilbert space $H$ gives the constant field $H_x=H$ , with measurability tested against a countable orthonormal basis. More generally, measurable subsets can carry fibers of different finite or countably infinite dimensions. Unlike a topological vector bundle, a measurable field need not have a topology on the disjoint union of its fibers.

References

M. Takesaki, Theory of Operator Algebras I, Springer, 1979. Springer DOI record. Relevant: Chapter IV, §8, and the chapter “Tensor Products of Operator Algebras and Direct Integrals.”