Direct integral of locally Hilbert spaces over a locally measure space

TL;DR

This paper generalizes the concept of direct integrals to locally Hilbert spaces over locally measure spaces, establishing their properties and the structure of associated operator algebras.

Contribution

It introduces the notion of direct integrals of locally Hilbert spaces over locally measure spaces and characterizes the structure of related operator algebras.

Findings

The direct integral of locally Hilbert spaces forms a locally Hilbert space.

Decomposable and diagonalizable operators form locally von Neumann algebras.

The algebra of diagonalizable operators is abelian and equals the commutant of decomposable operators.

Abstract

In this work, we introduce the concept of the direct integral of locally Hilbert spaces by generalizing the classical notion of a measure space to that of a locally measure space. We establish that the direct integral of a family of locally Hilbert spaces over a locally measure space forms a locally Hilbert space. We then define two important subclasses of locally bounded operators on such direct integrals, namely decomposable locally bounded operators and diagonalizable locally bounded operators. We show that each of these subclasses forms a locally von Neumann algebra, and in particular, that the locally von Neumann algebra of diagonalizable operators is abelian. Finally, we prove that the locally von Neumann algebra of diagonalizable operators coincides with the commutant of the locally von Neumann algebra of decomposable operators.