C${}^*$-algebras have been intensely studied in recent years, especially through the lens of K-theory, classification and the Eilliott program. Prominent advances in non-simple classification include an abundance of results for Cuntz-Krieger algebras of a directed graph. One such result of Cuntz and Krieger shows that the extension groups of such algebras coincide with Bowen-Franks groups of the subshift of finite type associated to the graph. A more recent achievement due to Eilers, Restorff, Ruiz and Sorensen is the complete classification of Cuntz-Krieger algebras of finite graphs up to a prescribed set of deformations of the graph arising from the subshift.

On the other hand, classifying non-self-adjoint algebras is an effort initiated by Arveson and Josephson in their late 60s paper on algebras arising from measure preserving dynamics. This was later taken up by many authors, including Davidson and Katsoulis who classified non-self-adjoint algebras arising from dynamical systems on topological spaces. In the late 90s, Muhly and Solel established a vastly applicable non-commutative function theory where completely contractive representations of tensor and Hardy algebras are treated as points. This framework allows us to treat many non-self-adjoint classification problems by modeling the algebras as tensor or Hardy algebras of C*-correspondences.

In this talk we will connect these seemingly unrelated topics, survey some pertinent results from the literature and uncover a striking hierarchy of classification for irreversible and reversible operator algebras.