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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 semigroupfunctor. Examples are given of algebras--simple and non-simple--for which thisclassification fails. It is shown that a suitable suspension of the Cuntzsemigroup functor deals successfully with some of these counterexamples.

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