Title: 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 the preservations 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.