Topological full groups arising from Cuntz and Cuntz-Toeplitz algebras and their crossed products
Ryoya Arimoto, Taro Sogabe
TL;DR
This work develops a groupoid–C*-algebraic framework for topological full groups arising from the Cuntz algebras $\mathcal{O}_n$ and Cuntz–Toeplitz algebras $\mathcal{E}_n$, and analyzes their crossed products with Cartan subalgebras. It provides explicit descriptions of abelianizations and normal subgroups for the Higman–Thompson-type groups $V_n$, $\Gamma_n$, and their infinite unions, establishing stability of the abelianization and a simple commutator in the infinite case. The authors determine the KMS and ground states for the natural $\mathbb{R}$-actions induced by cocycles on the associated transformation groupoids, showing existence and uniqueness results at sharp inverse temperatures (e.g., $\beta=\log n$) and relating these to traces on stabilizers via Neshveyev’s theory. The results illuminate how groupoid homology, C*-simplicity, and unique trace properties govern the equilibrium state structure of the reduced crossed products, and they connect the abelianization and simplicity facts to the spectral properties of the KMS spectrum. Overall, the paper advances understanding of the interplay between operator-algebraic invariants and dynamical properties of topological full groups from Cuntz-related groupoids.
Abstract
In this paper, we investigate the topological full groups arising from the Cuntz and Cuntz-Toeplitz algebras and their crossed products with the Cartan subalgebras of Cuntz and Cuntz-Toeplitz algebras. We study the normal subgroups and abelianization of these groups and completely determine the KMS states of the reduced crossed products with respect to some canonical gauge actions.