A Simple C∗-algebra with Finite Nuclear Dimension Which Is Not Z-stable

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 II1 factor not isomorphic to its tensor product with the separable hyperfinite II1 factor R. The proofs use a weakening of the Continuum Hypothesis. Elliott’s program of classification of nuclear (a.k.a. amenable) C∗-algebras recently underwent a transformative phase (see e.g., [5]). Following the counterexamples of Rørdam and Toms to the original program, it was realized that a regularity assumption stronger than the nuclearity is necessary for C∗-algebras to be classifiable by K-theoretic invariants. Conjecturally, regularity properties of three different flavours are all equivalent and are, modulo the UCT, sufficient for classification (restricting, say, to simple, separable, nuclear C∗-algebras). We shall consider two of these regularity assumptions on a C∗-algebra A. One of them asserts that A is Z-stable, meaning that it is isomorphic to its tensor product with the Jiang-Su algebra Z. An another postulates that A has finite nuclear dimension, this being a strengthening of the Completely Positive Approximation Property (CPAP) introduced by Winter and Zacharias in [18] (the CPAP is an equivalent formulation of amenability for C∗-algebras, see [3, Chapter 2]). The Toms– Winter conjecture states (among other things) that for separable, nuclear, simple, non-type I C∗-algebras, having finite nuclear dimension is equivalent to being Z-stable (see e.g., [17, 14]). The direct implication is a theorem of Winter ([16]). We show that it badly fails if the separability assumption is dropped. Theorem 1. The Continuum Hypothesis implies that there exists a simple nuclear C∗-algebra with nuclear dimension zero which is not Z-stable. The paradigm of regularity properties for C∗-algebras parallels certain older ideas in the study of von Neumann algebras. It has long been known that amenability for von Neumann algebras is equivalent to hyperfiniteness, and in the separable, non-type I case, it implies R-stability (the property of being isomorphic to one’s tensor product with the unique separable hyperfinite II1 factor R; von Neumann algebras with this property are commonly called McDuff) (see [13, Chapters XIV and XVI]). It is also not a stretch Dedicated to Stuart White on the occasion of his 33rd birthday.