The characterization of Z-stable graph algebras by regularity and graph conditions

Let $E$ be a row-finite graph, and let $C^*(E)$ be its graph $C^*$-algebra. An elementary subquotient is a subquotient isomorphic to a compact-operator algebra; purity is the regularity property used in the paper. Condition (K) and distinct detours are graph-theoretic properties.

Graph-algebra characterization conjecture. The following conditions are equivalent:

$$C^*(E) \text{ is } \text{-stable},$$ $$C^*(E) \text{ has no elementary subquotients},$$ $$C^*(E) \text{ is pure},$$ $$E \text{ has Condition (K) and distinct detours}.$$

The paper proves the corresponding equivalence for the graph-algebra results developed before the conjecture, including the relationship between no elementary subquotients, purity, Condition (K), and distinct detours. The full characterization for all row-finite graphs is proposed as an open problem.