The nuclear dimension of C*-algebras

TL;DR

This paper introduces the nuclear dimension for C*-algebras, a noncommutative analog of topological dimension, showing its stability, computability for key examples, and relevance to classification and coarse geometry.

Contribution

It defines the nuclear dimension, demonstrates its properties, and links it to classification theory and coarse geometry, expanding the understanding of C*-algebra structure.

Findings

Finite nuclear dimension for all UCT Kirchberg algebras.

Finite nuclear dimension in all classified nuclear C*-algebras.

Nuclear dimension relates to asymptotic dimension in coarse geometry.

Abstract

We introduce the nuclear dimension of a C*-algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive limits, tensor products, hereditary subalgebras (hence ideals), quotients, and even extensions. It can be computed for many examples; in particular, it is finite for all UCT Kirchberg algebras. In fact, all classes of nuclear C*-algebras which have so far been successfully classified consist of examples with finite nuclear dimension, and it turns out that finite nuclear dimension implies many properties relevant for the classification program. Surprisingly, the concept is also linked to coarse geometry, since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.