Rohklin dimension for C*-correspondences

Abstract

We extend the notion of Rokhlin dimension from topological dynamical systems to $C^*$-correspondences. We show that in the presence of finite Rokhlin dimension and a mild quasidiagonal-like condition (which, for example, is automatic for finitely generated projective correspondences), finite nuclear dimension passes from the scalar algebra to the associated Toeplitz--Pimsner and (hence) Cuntz--Pimsner algebras. As a consequence we provide new examples of classifiable $C^*$-algebras: if $A$ is simple, unital, has finite nuclear dimension and satisfies the UCT, then for every finitely generated projective $\mathcal{H}$ with finite Rokhlin dimension, the associated Cuntz--Pimsner algebra $\mathcal{O} (\mathcal{H})$ is classifiable in the sense of Elliott's Program.