Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras

TL;DR

We compute the quantum isometry groups of Cuntz--Krieger algebras $O_A$ equipped with a spectral triple from the log-Laplacian on the Deaconu--Renault groupoid of the underlying topological Markov chain. To organize the quantum symmetries, we introduce the Ariadne quantum groups $\mathbb{G}_A^{\ell}$ and prove that $\mathbb{G}_A^{\infty}$ is universal for $D$-isometric actions, i.e. the quantum isometry group of $O_A$. The analysis reveals rich noncommutative symmetry beyond the classical isometry group $\mathbb{T}\wr\mathrm{Aut}(A)$, including ergodic actions on the Cantor space and connections to quantum automorphism groups of graphs and easy quantum groups; specific instances show both termination (e.g. Fibonacci, where $\mathbb{G}_A^1=\mathbb{G}_A^{\infty}$) and non-termination (e.g. full shift, yielding unitary easy quantum groups). Overall, the work demonstrates that quantum symmetries of noncommutative spaces like $O_A$ are substantially richer than their classical counterparts and provide new interfaces between operator algebras, quantum groups, and noncommutative geometry.

Abstract

We compute the quantum isometry groups of Cuntz--Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain. This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups. Contrary to the classical isometry groups, whose actions on the Cuntz--Krieger algebras are never ergodic, the quantum isometry group acts ergodically in the case of the Cuntz algebra. This also leads to the construction of a (genuinely quantum) ergodic and faithful action of a compact matrix quantum group on the Cantor space.

Theorems & Definitions (67)

• Theorem A
• Theorem B
• Theorem C
• Remark 2.1
• Remark 2.2
• Remark 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Definition 3.4