Rohklin dimension for C*-correspondences

TL;DR

This paper generalizes Rokhlin dimension to $C^*$-correspondences and shows that under certain conditions, finite nuclear dimension is preserved in associated algebras, leading to new classifiable $C^*$-algebras.

Contribution

It introduces the concept of Rokhlin dimension for $C^*$-correspondences and demonstrates its implications for nuclear dimension and classification of related $C^*$-algebras.

Findings

Finite nuclear dimension passes from scalar algebra to Toeplitz--Pimsner and Cuntz--Pimsner algebras.

New examples of classifiable $C^*$-algebras are provided under the finite Rokhlin dimension condition.

The results apply to simple, unital $C^*$-algebras with finite nuclear dimension satisfying the UCT.

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.