Classification of C∗-homomorphisms from C0(0,1] to a C∗-algebra

A class of C∗-algebras is described for which the C∗-homomorphisms from C0(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.