Functional Models of Abelian Locally von Neumann Algebras and Direct Integrals of Locally Hilbert Spaces

TL;DR

This paper develops a functional model for Abelian locally von Neumann algebras acting on locally Hilbert spaces, characterizing them via locally essentially bounded functions and direct integrals.

Contribution

It introduces the concept of direct integrals of locally Hilbert spaces and characterizes Abelian locally von Neumann algebras acting on separable spaces.

Findings

Provides a functional model for Abelian locally von Neumann algebras.

Defines locally decomposable and locally diagonalisable operators.

Shows isomorphism between certain Abelian locally von Neumann algebras and diagonalisable operators.

Abstract

We obtain a functional model for an arbitrary Abelian locally von Neumann algebra acting on a representing locally Hilbert space under the assumption that the index directed set is countable, in terms of locally essentially bounded functions on strictly inductive systems of measure spaces, which can be viewed as the reduction theory of this kind of operator algebras. Then, we single out the concept of a direct integral of locally Hilbert spaces and the concepts of locally decomposable and locally diagonlisable operators and we show that these form locally von Neumann algebras that are commutant one to each other. Finally, we show that any Abelian locally von Neumann algebra, which acts on separable representing locally Hilbert spaces and such that the index set is a sequentially finite directed set, is spatially isomorphic with the Abelian locally von Neumann algebra of all locally diagonlisable operators on a certain direct integral of locally Hilbert spaces with respect to a certain strictly inductive system of locally finite measure spaces on standard Borel spaces.