Shift Equivalence Through the Lens of Cuntz–Krieger Algebras

The paper proves that A and B are shift equivalent iff their Cuntz–Krieger algebras are gauge-equivariantly stably isomorphic (see the abstract and Corollary 5.7, which equates (1) shift equivalence with (2) stably equivariant isomorphism) . In contrast, the model’s forward direction relies on two unsupported steps: (i) that strong shift equivalence of certain correspondences automatically yields a gauge-equivariant stable isomorphism of the associated Cuntz–Pimsner algebras, which the paper explicitly notes is not known in general and must be circumvented ; and (ii) that O(E) is stably isomorphic (gauge-equivariantly) to O(E^{⊗ m}), which is not established. The reverse (⇐) direction of the model broadly aligns with the paper’s K-theoretic route via dimension triples, but the overall proof is incomplete/incorrect in the forward implication.