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

Toke Meier Carlsen

doi:10.1016/j.aim.2017.11.018

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

Rings and Algebras 2018-04-12 v3 Operator Algebras

Authors: Toke Meier Carlsen

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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 $\mathbb{Z}$ .

Keywords

Leavitt path algebras *-isomorphism countable directed graphs graph groupoids graph C*-algebras diagonal preserving subrings of C

Cite

@article{arxiv.1601.00777,
  title  = {$*$-isomorphism of Leavitt path algebras over $\mathbb{Z}$},
  author = {Toke Meier Carlsen},
  journal= {arXiv preprint arXiv:1601.00777},
  year   = {2018}
}

Comments

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

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

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