Ranks of operators in simple C*-algebras with stable rank one

TL;DR

This paper demonstrates that in certain simple C*-algebras with stable rank one, every suitable affine function corresponds to an operator's rank, and establishes conditions for $ ext{Z}$-stability related to the Toms-Winter conjecture.

Contribution

It proves the realization of affine functions as operator ranks in stable rank one C*-algebras and links $ ext{Z}$-stability to strict comparison under finite nuclear dimension.

Findings

Every affine function on the quasitrace simplex is realized as an operator rank.

$ ext{Z}$-stability is equivalent to strict comparison in these algebras.

The Toms-Winter conjecture holds for a class of approximately subhomogeneous C*-algebras.

Abstract

Let $A$ be a separable, unital, 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 separable, unital, simple, approximately subhomogeneous C*-algebras with stable rank one.