Nuclear dimension and n-comparison

TL;DR

The paper demonstrates that C*-algebras with finite nuclear dimension have the n-comparison property in their Cuntz semigroup, leading to criteria for stability based on unital quotients and traces.

Contribution

It establishes a link between nuclear dimension and n-comparison in the Cuntz semigroup, extending stability criteria for C*-algebras.

Findings

Finite nuclear dimension implies n-comparison in the Cuntz semigroup.

Finite nuclear dimension C*-algebras are stable iff they lack non-zero unital quotients and bounded traces.

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 $\sigma$-unital C*-algebras of finite nuclear dimension (and even of nuclear dimension at most $\omega$) are stable if and only if they have no non-zero unital quotients and no non-zero bounded traces.