A dynamical characterization of diagonal preserving $\ast$-isomorphisms of graph $C^{\ast }$-algebras

Abstract

We characterize when there exists a diagonal preserving $\ast$-isomorphism between two graph $C^{\ast }$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of "orbit equivalence" between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal preserving $\ast$-isomorphism between the graph $C^{\ast }$-algebras $C^{\ast }(E)$ and $C^{\ast }(F)$.

References in corpus (3)

• Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras
• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Geometric classification of graph $C^{\ast }$-algebras over finite graphs

Cited by in corpus (6)

• Diagonal-preserving gauge-invariant isomorphisms of graph $C^{\ast }$-algebras
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• Orbit equivalence of graphs and isomorphism of graph groupoids
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• Orbit equivalence of higher-rank graphs
• State splitting, strong shift equivalence and stable isomorphism of Cuntz-Krieger algebras