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

Stefaan Vaes, Lise Wouters

TL;DR

The paper develops a comprehensive, rigorous framework for Borel and measured fields of separable structures—Polish spaces, Banach spaces, C$^*$-algebras, and von Neumann algebras—over standard Borel bases. It establishes equivalences between abstract and concrete realizations, proves stability of constructions (duals, completions, tensor and crossed products), and provides both positive results (e.g., dual balls, L^p-fields, and dualizable structures) and sharp counterexamples (notably for automorphism fields). A key contribution is a suite of universal Borel fields that codify separable structures for robust Borel-analytic and complexity questions. The framework acts as a practical toolbox for operator-algebraic contexts requiring measurable fields, with wide-reaching implications for Borel coding and classification problems.

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.

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

TL;DR

The paper develops a comprehensive, rigorous framework for Borel and measured fields of separable structures—Polish spaces, Banach spaces, C-algebras, and von Neumann algebras—over standard Borel bases. It establishes equivalences between abstract and concrete realizations, proves stability of constructions (duals, completions, tensor and crossed products), and provides both positive results (e.g., dual balls, L^p-fields, and dualizable structures) and sharp counterexamples (notably for automorphism fields). A key contribution is a suite of universal Borel fields that codify separable structures for robust Borel-analytic and complexity questions. The framework acts as a practical toolbox for operator-algebraic contexts requiring measurable fields, with wide-reaching implications for Borel coding and classification problems.

Abstract

Several recent articles in operator algebras make a nontrivial use of the theory of measurable fields of von Neumann algebras and related structures. This includes the associated field 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 of von Neumann algebras, the field of Polish groups need not be Borel.
Paper Structure (12 sections, 49 theorems, 145 equations)

This paper contains 12 sections, 49 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Borel and measured fields of Polish spaces
  3. Closed subfields and relation with the Effros Borel structure
  4. Construction and uniqueness of standard Borel spaces
  5. Borel and measured fields of separable Banach spaces and duality
  6. Borel fields of separable metric spaces, normed spaces and completions
  7. Borel fields of bounded operators
  8. Borel fields of von Neumann algebras: abstract vs. concrete
  9. Borel fields of automorphism groups: a counterexample
  10. Borel fields of second countable locally compact spaces and groups
  11. Tensor products and crossed products
  12. Universal Borel fields

Key Result

Lemma 2.3

Let $V = (V_x)_{x \in X}$ be a Borel field of Polish spaces and let $d$ be a compatible Borel metric.

Theorems & Definitions (108)

  • Definition 2.1
  • Example 2.2: Sri98
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • Proposition 3.1
  • ...and 98 more