A simple C*-algebra with finite nuclear dimension which is not Z-stable

TL;DR

This paper constructs a simple C*-algebra with finite nuclear dimension that does not exhibit Z-stability, challenging assumptions in the classification of nuclear C*-algebras.

Contribution

It provides the first example of a simple C*-algebra with finite nuclear dimension that is not Z-stable, using a weakened form of the Continuum Hypothesis.

Findings

Existence of a simple C*-algebra with nuclear dimension zero not Z-stable

Construction of a hyperfinite II_1 factor not tensorially isomorphic to R

Use of a weakened Continuum Hypothesis in proofs

Abstract

We construct a simple C*-algebra with nuclear dimension zero that is not isomorphic to its tensor product with the Jiang-Su algebra Z, and a hyperfinite II_1 factor not isomorphic to its tensor product with the separable hyperfinite II_1 factor R. The proofs use a weakening of the Continuum Hypothesis.