$\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$

Abstract

We characterise when the Leavitt path algebras over $\mathbb{Z}$ of two arbitrary countable directed graphs are $\ast$-isomorphic by showing that two Leavitt path algebras over $\mathbb{Z}$ are $\ast$-isomorphic if and only if the corresponding graph groupoids are isomorphic (if and only if there is a diagonal preserving isomorphism between the corresponding graph $C^{\ast }$-algebras). We also prove that any $\ast$-homomorphism between two Leavitt path algebras over $\mathbb{Z}$ maps the diagonal to the diagonal. Both results hold for slight more general subrings of $\mathbb{C}$ than just $\mathbb{Z}$.

9 pp. Thm 1 and Cor 5 have been changed to emphasize that the *-homomorphisms are *-algebra homomorphisms. The references [6], [10], [12] and [21] have been added, and the remarks following Thm 1 have been updated in order to reflect new results in these papers. There is no longer any Remark 4, so Prop 5 has become Prop 4, and Cor 6 has become Cor 5. This is the version that will be published

References in corpus (3)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Orbit equivalence of graphs and isomorphism of graph groupoids
• A dynamical characterization of diagonal preserving $\ast$-isomorphisms of graph $C^{\ast }$-algebras

Cited by in corpus (10)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• The complete classification of unital graph $C^{\ast }$-algebras: Geometric and strong
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• Orbit equivalence of graphs and isomorphism of graph groupoids
• Graded C*-algebras and twisted groupoid C*-algebras
• Reconstruction of twisted Steinberg algebras
• Lifting morphisms between graded Grothendieck groups of Leavitt path algebras
• Classification conjectures for Leavitt path algebras
• Tensor products of Steinberg algebras
• A note on the core of Steinberg algebras