Generators of von Neumann algebras associated with spectral measures

TL;DR

This paper characterizes all generators of von Neumann algebras associated with spectral measures, including unbounded generators, using integrals with respect to the spectral measure.

Contribution

It provides a simple description of all generator sets for von Neumann algebras linked to spectral measures, extending to unbounded affiliated generators.

Findings

Complete characterization of generator sets for von Neumann algebras from spectral measures

Includes description of unbounded generators affiliated with the algebra

Simplifies understanding of algebra generation via spectral integrals

Abstract

Let $P_E$ be the set of all values of a spectral measure $E$ and $A(P_E)$ be the smallest von Neumann algebra containing $P_E$. We give a simple description of all sets of generators of $A(P_E)$ in terms of the integrals with respect to $E$. The treatment covers not only the case of generators belonging to $A(P_E)$, but also the case of (possibly unbounded) generators affiliated with this algebra.