Model theory and Rokhlin dimension for compact quantum group actions

Abstract

We show that, for a given compact or discrete quantum group $G$, the class of actions of $G$ on C\*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for $G$-C\*-algebra, introduced by Barlak, Szabó, and Voigt in the case when $G$ is second countable and coexact, to an arbitrary compact quantum group $G$. All the preservation and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant \*-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C\*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.