Hazrat’s conjecture on the equivalence between shift equivalence, gauge-preserving stable isomorphism, and graded Morita equivalence

Establish the equivalence of the following three conditions for any pair of finite essential matrices A and B: (i) A and B are shift equivalent; (ii) the Cuntz–Krieger graph C*-algebras O_A and O_B are stably isomorphic via a *-isomorphism that preserves the canonical gauge actions γ_A and γ_B; and (iii) the Leavitt path algebras L_A and L_B are graded Morita equivalent.

Background

The paper discusses the classification of subshifts of finite type via Cuntz–Krieger graph C*-algebras and their algebraic counterparts, Leavitt path algebras. Krieger’s corollary establishes that gauge-preserving stable isomorphism of Cuntz–Krieger graph C*-algebras implies shift equivalence of the corresponding matrices. Hazrat formulated a conjecture proposing a full equivalence among shift equivalence, gauge-preserving stable isomorphism of the associated Cuntz–Krieger graph C*-algebras, and graded Morita equivalence of the Leavitt path algebras.

This conjecture is central to bridging C*-algebraic and purely algebraic frameworks for classifying dynamical systems arising from adjacency matrices. The present paper provides homotopy-theoretic results that may support progress toward the conjecture but does not resolve it in full.