The nuclear dimension of [formula omitted]-stable C⁎-algebras

Abstract

We show that every nuclear O∞-stable ⁎-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, O∞-stable C⁎-algebras have nuclear dimension 1. We also characterise when O∞-stable C⁎-algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full O2-stable ⁎-homomorphisms have nuclear dimension 0.