The Universal Theory Of The Hyperfinite II$_1$ Factor Is Not Computable

TL;DR

This paper proves that the universal theory of the hyperfinite II$_1$ factor cannot be computed, using recent results linking it to the MIP*=RE problem, and provides a negative solution to the Connes Embedding Problem.

Contribution

It establishes the non-computability of the universal theory of the hyperfinite II$_1$ factor, connecting it to major unresolved problems in operator algebras.

Findings

The universal theory of the hyperfinite II$_1$ factor is not computable.

The proof leverages the MIP*=RE result.

The Connes Embedding Problem has a negative solution.

Abstract

We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg's QWEP Conjecture and Tsirelson's Problem.+