Diagonal-preserving gauge-invariant isomorphisms of graph $C^*$-algebras

TL;DR

This paper characterizes when graph $C^*$-algebras are isomorphic via diagonal-preserving maps that respect gauge actions, linking algebraic isomorphisms to dynamical properties of associated subshifts.

Contribution

It provides a groupoid-based characterization of diagonal-preserving gauge-invariant isomorphisms for graph $C^*$-algebras, connecting algebraic and dynamical conjugacy.

Findings

Isomorphism of Cuntz-Krieger algebras with gauge-intertwining maps corresponds to eventual conjugacy of subshifts.

Stabilized Cuntz-Krieger algebras are isomorphic with gauge-intertwining maps iff the subshifts are conjugate.

The paper links algebraic isomorphisms to dynamical conjugacy through groupoid cocycles.

Abstract

We study graph $C^*$-algebras equipped with generalised gauge actions, and characterise in terms of groupoids and groupoid cocycles when two graph $C^*$-algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the generalised gauge actions. We apply this characterisation to show that two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding one-sided subshifts are eventually conjugate, and that the stabilisation of two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding two-sided subshifts are conjugate.