Principal Groupoid Models for Cuntz Algebras and their Dynamic Asymptotic Dimension

TL;DR

This work computes the dynamic asymptotic dimension of principal Deaconu–Renault groupoid models for Cuntz algebras and shows that dad=1 in a broad class, enabling the construction of infinitely many non-conjugate C*-diagonals with Cantor spectrum for O2. By exploiting UHF-stability and KK-type results, the authors extend the construction to other O_k, obtaining infinitely many diagonals for even k and at least two non-isomorphic models for odd k. The approach highlights dynamic asymptotic dimension as a distinguishing invariant among groupoid models of Cuntz algebras and provides a framework for generating rich diagonal structures via groupoid techniques. These results have potential implications for the classification of C*-diagonals and the broader understanding of principal étale groupoid models in the Kirchberg–Phillips regime.

Abstract

We compute the dynamic asymptotic dimension of the principal groupoid models for the Cuntz algebras $\mathcal{O}_k$ for $2 \leq k < \infty$ that have arisen from work of Winter and the authors. Our method generalises to a wide class of Deaconu-Renault groupoids. As an application of our results, we prove that $\mathcal{O}_2$ has infinitely many non-conjugate C$^*$-diagonals with Cantor spectrum, and we generalise this result to other Cuntz algebras by combining the main result with work of Kopsacheilis-Winter and Brown-Clark-Sierakowski-Sims.

Theorems & Definitions (49)

• Theorem A
• Theorem B
• Theorem C
• Definition 1: cf. renault2008cartan
• Theorem 2: kumjian1986crenault2008cartan
• Definition 3
• Lemma 4
• proof
• Proposition 5
• proof