Principal Groupoid Models for Cuntz Algebras and their Dynamic Asymptotic Dimension

TL;DR

This paper calculates the dynamic asymptotic dimension of principal groupoid models for Cuntz algebras, revealing new structural properties and classifying diagonals with Cantor spectrum.

Contribution

It introduces a method to compute dynamic asymptotic dimension for a broad class of Deaconu-Renault groupoids, extending understanding of Cuntz algebra models.

Findings

Computed dynamic asymptotic dimension for principal groupoid models of Cuntz algebras.

Proved existence of infinitely many non-conjugate C*-diagonals with Cantor spectrum in $\\mathcal{O}_2$.

Generalized results to other Cuntz algebras using existing work.

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.