Nuclear dimension and n-comparison

Abstract

It is shown that if a C*-algebra has nuclear dimension $n$ then its Cuntz semigroup has the property of $n$-comparison. It then follows from results by Ortega, Perera, and Rordam that $σ$-unital C*-algebras of finite nuclear dimension (and even of nuclear dimension at most $ω$) are stable if and only if they have no non-zero unital quotients and no non-zero bounded traces.