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

Hannes Thiel

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arXiv · 1711.04721

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

Hannes Thiel

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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.

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BibTeX RIS

Hannes Thiel. 2017-11-13. Ranks of operators in simple C*-algebras with stable rank one. https://arxiv.org/abs/1711.04721

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