Nuclear dimension and Z-stability of pure C*-algebras

Abstract: In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially) (m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) \bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension. The main result says that if a separable, simple, nonelementary, unital C*-algebra A with locally finite nuclear dimension is (m,\bar{m})-pure, then it absorbs the Jiang-Su algebra Z tensorially. It follows that A is Z-stable if and only if it has the Cuntz semigroup of a Z-stable C*-algebra. The result may be regarded as a version of Kirchberg's celebrated theorem that separable, simple, nuclear, purely infinite C*-algebras absorb the Cuntz algebra O_\infty tensorially. As a corollary we obtain that finite nuclear dimension implies Z-stability for separable, simple, nonelementary, unital C*-algebras; this settles an important case of a conjecture by Toms and the author. The main result also has a number of consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. In particular, it completes the classification of simple, unital, approximately homogeneous algebras with slow dimension growth by their Elliott invariants, a question left open in the Elliott-Gong-Li classification of simple AH algebras. Another consequence is that for simple, unital, approximately subhomogeneous algebras, slow dimension growth and Z-stability are equivalent. In the case where projections separate traces, this completes the classification of simple, unital, approximately subhomogeneous algebras with slow dimension growth by their ordered K-groups.