Geometric Structure of Dimension Functions of Certain Continuous Fields

TL;DR

This paper investigates the geometric structure of dimension functions in continuous fields of C*-algebras, providing insights into their Cuntz semigroup and functional properties, and resolving a conjecture by Blackadar and Handelman.

Contribution

It offers new structural results on the Cuntz semigroup of continuous fields and characterizes when their stable rank is one, addressing longstanding conjectures.

Findings

Resolved a conjecture by Blackadar and Handelman.

Determined conditions for stable rank one in continuous fields over one-dimensional spaces.

Analyzed the geometric structure of dimension functions in these fields.

Abstract

In this paper we study structural properties of the Cuntz semigroup and its functionals for continuous fields of C*-algebras over finite dimensional spaces. In a variety of cases, this leads to an answer to a conjecture posed by Blackadar and Handelman. Enroute to our results, we determine when the stable rank of continuous fields of C*-algebras over one dimensional spaces is one.