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The Universal Theory of Locally Universal Tracial von Neumann Algebras is not Computable

Jananan Arulseelan, Aareyan Manzoor

Abstract

Building on Lin's breakthrough MIP$^{co}$ = coRE and an encoding of non-local games as universal sentences in the language of tracial von Neumann algebras, we show that locally universal tracial von Neumann algebras have undecidable universal theories. This implies that no such algebra admits a computable presentation. Our results also provide, for the first time, explicit examples of separable II$_1$ factors without computable presentations, and in fact yield a broad family of them, including McDuff factors, factors without property Gamma, and property (T) factors. We also obtain analogous results for locally universal semifinite von Neumann algebras and tracial C*-algebras. The latter provides strong evidence for a negative solution to the Kirchberg Embedding Problem. We discuss how these are obstructions to approximation properties in the class of tracial and semifinite von Neumann algebras.

The Universal Theory of Locally Universal Tracial von Neumann Algebras is not Computable

Abstract

Building on Lin's breakthrough MIP = coRE and an encoding of non-local games as universal sentences in the language of tracial von Neumann algebras, we show that locally universal tracial von Neumann algebras have undecidable universal theories. This implies that no such algebra admits a computable presentation. Our results also provide, for the first time, explicit examples of separable II factors without computable presentations, and in fact yield a broad family of them, including McDuff factors, factors without property Gamma, and property (T) factors. We also obtain analogous results for locally universal semifinite von Neumann algebras and tracial C*-algebras. The latter provides strong evidence for a negative solution to the Kirchberg Embedding Problem. We discuss how these are obstructions to approximation properties in the class of tracial and semifinite von Neumann algebras.

Paper Structure

This paper contains 9 sections, 14 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. MIPco = coRE and its Consequences
  4. Non-local Games
  5. Universal Sentences for Game Values
  6. Applications to Computable Continuous Model Theory
  7. Locally Universal Models
  8. Results on Computability of Universal Theories
  9. Consequences for C*-algebras

Key Result

Theorem 1

Suppose $\mathsf{MIP}^{\mathsf{co}}=\mathsf{coRE}$, then for each Turing machine $M$, there is a restricted universal sentence $\sigma_M$ in the language of tracial von Neumann algebras, computable from the description of $M$, such that: Here the supremum ranges over all separable tracial von Neumann algebras $\mathcal{N}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (27)

  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 17 more