Classification of inductive limits of 1-dimensional NCCW complexes

TL;DR

This paper classifies a specific class of 1-dimensional noncommutative C*-algebras using a new functor based on the Cuntz semigroup, and applies this to show isomorphisms of certain crossed product algebras.

Contribution

It introduces a classification framework for inductive limits of 1-dimensional NCCW complexes with trivial K_1, using the Cu functor and simplifies it for simple cases.

Findings

Classification of certain 1D NCCW C*-algebras achieved.

Cu functor reduces to K_0, traces, and pairing for simple algebras.

Crossed products by quasi-free actions are all isomorphic for dense irrationals.

Abstract

A classification result is obtained for the C*-algebras that are (stably isomorphic to) inductive limits of 1-dimensional noncommutative CW complexes with trivial $K_1$-group. The classifying functor Cu is defined in terms of the Cuntz semigroup of the unitization of the algebra. For the simple C*-algebras covered by the classification, Cu reduces to the ordered $K_0$-group, the cone of traces, and the pairing between them. As an application of the classification, it is shown that the crossed products by a quasi-free action $O_2\rtimes_\lambda \mathbb{R}$ are all isomorphic for a dense set of positive irrational numbers $\lambda$.