Ranks of Operators in Simple $$C^*$$-algebras with Stable Rank One

Abstract

Let A be a simple $$C^*$$-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of A is realized as the rank of an operator in the stabilization of A. Assuming moreover that A has locally finite nuclear dimension, we deduce that A is $$\mathcal {Z}$$-stable if and only if it has strict comparison of positive elements. In particular, the Toms–Winter conjecture holds for simple, approximately subhomogeneous $$C^*$$-algebras with stable rank one.