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

TL;DR

This paper characterizes when Leavitt path algebras over $\\mathbb{Z}$ are $*$-isomorphic by relating algebraic isomorphisms to graph groupoid isomorphisms and diagonal-preserving $C^*$-algebra isomorphisms.

Contribution

It establishes a complete characterization of $*$-isomorphisms of Leavitt path algebras over $\\mathbb{Z}$ via graph groupoid and $C^*$-algebra isomorphisms, extending to certain subrings of $\mathbb{C}$.

Findings

Leavitt path algebras over $\\mathbb{Z}$ are $*$-isomorphic iff their graph groupoids are isomorphic.

Any $*$-homomorphism between such algebras preserves the diagonal.

Results extend to some subrings of $\mathbb{C}$.

Abstract

We characterise when the Leavitt path algebras over $\mathbb{Z}$ of two arbitrary countable directed graphs are $*$-isomorphic by showing that two Leavitt path algebras over $\mathbb{Z}$ are $*$-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 $C^*$-algebras). We also prove that any $*$-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 $\mathbb{Z}$.