Borel fields and measured fields of Polish spaces, Banach spaces, von Neumann algebras and C*-algebras

TL;DR

This paper develops a rigorous and flexible theory of Borel and measurable fields of operator algebras, addressing gaps in the literature and illustrating subtle issues with counterexamples.

Contribution

It provides a comprehensive framework for Borel fields of von Neumann algebras and related structures, including counterexamples highlighting subtle measurability issues.

Findings

Counterexamples show automorphism groups may not form Borel fields

Established a rigorous theory for measurable fields of operator algebras

Filled a gap in the literature on Borel and measurable fields

Abstract

Several recent articles in operator algebras make a nontrivial use of the theory of measurable fields of von Neumann algebras $(M_x)_{x \in X}$ and related structures. This includes the associated field $(\text{Aut}\ M_x)_{x \in X}$ of automorphism groups and more general measurable fields of Polish groups with actions on Polish spaces. Nevertheless, a fully rigorous and at the same time sufficiently broad and flexible theory of such Borel fields and measurable fields is not available in the literature. We fill this gap in this paper and include a few counterexamples to illustrate the subtlety: for instance, for a Borel field $(M_x)_{x \in X}$ of von Neumann algebras, the field of Polish groups $(\text{Aut}\ M_x)_{x \in X}$ need not be Borel.