Nuclear dimension of groupoid C*-algebras with large abelian isotropy, with applications to C*-algebras of directed graphs and twists

TL;DR

This paper characterizes when groupoid C*-algebras are subhomogeneous and provides bounds on their nuclear dimension, with applications to graph C*-algebras and twisted groupoid C*-algebras, extending previous results to cases with large isotropy.

Contribution

It generalizes nuclear dimension bounds to groupoids with large isotropy, including graph groupoids, and establishes new bounds for twisted groupoid C*-algebras.

Findings

All stably finite graph C*-algebras have nuclear dimension at most 1.

Bound on nuclear dimension depends on topological and spectral properties of the groupoid.

Nuclear dimension bounds extend to twisted groupoid C*-algebras.

Abstract

We characterise when the C*-algebra C*(G) of a locally compact and Hausdorff groupoid G is subhomogeneous, that is, when its irreducible representations have bounded finite dimension; if so we establish a bound for its nuclear dimension in terms of the topological dimensions of the unit space of the groupoid and the spectra of the primitive ideal spaces of the isotropy subgroups. For an etale groupoid G, we also establish a bound on the nuclear dimension of its C*-algebra provided the quotient of G by its isotropy subgroupid has finite dynamic asymptotic dimension in the sense of Guentner, Willet and Yu. Our results generalise those of C. B\"oncicke and K. Li to groupoids with large isotropy, including graph groupoids of directed graphs. We find that all graph C*-algebras that are stably finite have nuclear dimension at most 1. We also show that the nuclear dimension of the C*-algebra of a twist over G has the same bound on the nuclear dimension as for C*(G) and the twisted groupoid C*-algebra.