Complexity rank one implies real rank zero

Qingnan An, Zhichao Liu

arXiv:2608.12089v2 Primary: math.OA

Abstract

We show that $C^*$-algebras with complexity rank one have real rank zero, as an application, the uniform Roe algebra $C_u^*|\mathbb{Z}|$ has real rank zero.

Disclosure

“15 It is also used for language editing. All arguments and ideas are developed and substantially revised by the authors with the exception of preliminary drafts of the proofs of Lemma 2.7 which were developed with the help of ChatGPT. The authors take full responsibility of the content of this paper. References [1] M. Amini, K. Li, D. Sawicki and A. Shakibazadeh. Dynamic asymptotic dimension for actions of virtually cyclic groups.”
PDF page 15

Classification: Drafting a complete proof for author revision
Multiplier: 9
Verified: 2026-08-23T05:11:47.154453Z

Structural counts

Pages 16 pdf

Theorems 3 source

Lemmas 5 source

Propositions 3 source

Corollaries 3 source

Definitions 8 source

Displayed equations 94 source

Bibliography entries 24 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file complexrank.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.