The complete classification of unital graph $C^*$-algebras: Geometric and strong

TL;DR

This paper provides a complete geometric classification of unital graph $C^*$-algebras, showing Morita equivalence is characterized by filtered $K$-theory and realized through graph moves, making the classification decidable.

Contribution

It introduces new graph moves that, combined with existing invariants, fully classify unital graph $C^*$-algebras up to Morita equivalence.

Findings

Morita equivalence is determined by ordered, filtered $K$-theory.

New graph moves leave the algebra invariant and complete the classification.

Every $K$-theory isomorphism lifts to a $*$-isomorphism of the algebras.

Abstract

We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered $K$-theory. The classification result is geometric in the sense that it establishes that any Morita equivalence between $C^*(E)$ and $C^*(F)$ in this class can be realized by a sequence of moves leading from $E$ to $F$, in a way resembling the role of Reidemeister moves on knots. As a key ingredient, we introduce a new class of such moves, establish that they leave the graph algebras invariant, and prove that after this augmentation, the list of moves becomes complete in the sense described above. Along the way, we prove that every ordered, reduced filtered $K$-theory isomorphism can be lifted to an isomorphism between the stabilized $C^*$-algebras - and, as a consequence, that every ordered, reduced filtered $K$-theory isomorphism preserving the class of the unit comes from a $*$-isomorphism between the unital graph $C^*$-algebras themselves. It follows that the question of Morita equivalence amongst unital graph $C^*$-algebras is a decidable one. As immediate examples of applications of our results we revisit the classification problem for quantum lens spaces and verify, in the unital case, the Abrams-Tomforde conjectures.