Towards a classification of unitary elements of C*-algebras

In [5] the author conjectures and partially shows that the Cuntz semigroup classifies unitary elements of unital AF-algebras. We provide a complete proof by addressing the existence part of the conjecture, under a mild adjustment of both domain and codomain of the functor Cu. We also tackle the classification beyond the AF case and more particularly, we look at unitary elements of what we call AH${}_1$-algebras. We obtain positive progress as far as the existence part is concerned. Nevertheless, we reveal that extra information is needed for the uniqueness part of the classification that the Cuntz semigroup fails to capture.