Nuclear Dimension of Groupoid C*-algebras with Large Abelian Isotropy, With Applications to C*-algebras of Directed Graphs and Twists

Abstract We characterise when the $C^{\ast }$-algebra $C^{\ast }(G)$ 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 étale groupoid $G$, we also establish a bound on the nuclear dimension of its $C^{\ast }$-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. Böncicke and K. Li to groupoids with large isotropy, including graph groupoids of directed graphs. We find that all graph $C^{\ast }$-algebras that are stably finite have nuclear dimension at most $1$. We also show that the nuclear dimension of the $C^{\ast }$-algebra of a twist over $G$ has the same bound on the nuclear dimension as for $C^{\ast }(G)$ and the twisted groupoid $C^{\ast }$-algebra.