The diagonal dimension of sub-C*-algebras

TL;DR

This paper introduces diagonal dimension, a new invariant for diagonal sub-C*-algebras that captures refined dynamical information and relates to existing dimensions like nuclear and asymptotic dimension.

Contribution

It defines diagonal dimension, proves its key properties, and establishes its connections with dynamical and geometric dimensions in various contexts.

Findings

Diagonal dimension detects more refined information than nuclear dimension.

Diagonal dimension of crossed products relates to tower and covering dimensions.

For groupoid C*-algebras, diagonal dimension bounds asymptotic dimension.

Abstract

We introduce diagonal dimension, a version of nuclear dimension for diagonal sub-C*-algebras (sometimes also referred to as diagonal C*-pairs). Our concept has good permanence properties and detects more refined information than nuclear dimension. In many situations it is precisely how dynamical information is encoded in an associated C*-pair. For free actions on compact Hausdorff spaces, diagonal dimension of the crossed product with its canonical diagonal is bounded above by a product involving Kerr's tower dimension of the action and covering dimension of the space. It is bounded below by the dimension of the space, by the asymptotic dimension of the group, and by the fine tower dimension of the action. For a locally compact, Hausdorff, \'etale groupoid, diagonal dimension of the groupoid C*-algebra is bounded below by the dynamic asymptotic dimension of the groupoid. For free Cantor dynamical systems, diagonal dimension (defined at the level of the crossed product C*-algebra) and tower dimension (an entirely dynamical notion) agree on the nose. Similarly, for a finitely generated group diagonal dimension of its uniform Roe algebra with the canonical diagonal agrees precisely with asymptotic dimension of the group. This statement also holds for uniformly bounded metric spaces. We apply the lower bounds above to a number of further examples which show how diagonal dimension keeps track of information not seen by nuclear dimension.