Nuclear dimension and Z-stability

Simple, separable, unital, monotracial and nuclear C∗-algebras are shown to have finite nuclear dimension whenever they absorb the Jiang–Su algebraZ tensorially. This completes the proof of the Toms–Winter conjecture in the unique trace case. The structure theory of simple nuclearC∗-algebras is currently undergoing revolutionary progress, driven by the discovery of regularity properties of various flavours: topological, functional analytic and algebraic. Despite the diverse nature of these regularity properties, they are all satisfied by those classes of C∗-algebras which have been successfully classified by K -theoretic data, and they all fail spectacularly for the “exotic” algebras in [30,40] which provide counterexamples to Elliott’s classification conjecture. The observation that Research partially supported by JSPS (the Grant-in-Aid for Research Activity Start-up 25887031), by EPSRC (Grant no. I019227/1-2), and by the DFG (SFB 878). Y. Sato Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan e-mail: [email protected] S. White School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow Q12 8QW, Scotland e-mail: [email protected] W. Winter (B) Mathematisches Institut der WWU Münster, Einsteinstraße 62, 48149 Münster, Germany e-mail: [email protected]