Direct integral of locally Hilbert spaces

TL;DR

This paper introduces the concept of the direct integral of locally Hilbert spaces, establishing their structure and properties, including classes of bounded operators and their algebraic relations.

Contribution

It formalizes the direct integral of locally Hilbert spaces and characterizes associated operator classes as von Neumann algebras, advancing the mathematical framework.

Findings

Decomposable operators form a locally von Neumann algebra

Diagonalizable operators form an abelian von Neumann algebra

These two classes are mutual commutants

Abstract

In this work, we introduce the concept of the direct integral of locally Hilbert spaces. This notion is formulated such that the direct integral of locally Hilbert spaces forms a locally Hilbert space. We then define two classes of locally bounded operators on the direct integral of locally Hilbert spaces namely the class of decomposable locally bounded operators and the class of diagonalizable locally bounded operators. We prove that the set of all decomposable locally bounded operators forms a locally von Neumann algebra, while the set of diagonalizable locally bounded operators forms an abelian von Neumann algebra, and we show that they are commutant of each other.