Classification of homomorphisms from $C_0(0,1]$ to a C*-algebra

TL;DR

This paper investigates the classification of homomorphisms from $C_0(0,1]$ to C*-algebras using the Cuntz semigroup, identifying cases where the classification applies and proposing a suspension method to address counterexamples.

Contribution

It introduces a classification framework for certain C*-algebras via the Cuntz semigroup and proposes a suspension technique to handle previously unresolved cases.

Findings

Classification applies to a specific class of C*-algebras.

Counterexamples are identified where classification fails.

Suspension of the Cuntz semigroup helps classify some counterexamples.

Abstract

A class of C*-algebras is described for which the homomorphism from $C_0(0,1]$ to the algebra may be classified by means of the Cuntz semigroup functor. Examples are given of algebras--simple and non-simple--for which this classification fails. It is shown that a suitable suspension of the Cuntz semigroup functor deals successfully with some of these counterexamples.