Towards a classification of unitary elements of C*-algebras

TL;DR

This paper proves that the Cuntz semigroup classifies unitary elements in unital AF-algebras and explores classification challenges for more general AH$_1$-algebras, highlighting both progress and limitations.

Contribution

It provides a complete proof for the classification of unitaries in AF-algebras and investigates the classification beyond AF-algebras, identifying the need for additional invariants.

Findings

Complete proof of classification for AF-algebras

Progress in classifying unitaries in AH$_1$-algebras

Identification of limitations of the Cuntz semigroup for uniqueness

Abstract

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.