The Cuntz splice does not preserve $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$

Abstract

We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $\ast$-algebras. There are two key ingredients in the proof. One is a partial algebraic translation of Matsumoto and Matui's result on diagonal preserving isomorphisms of Cuntz--Krieger algebras. The other is a complete description of the projections in $L_{\mathbb{Z}}(E)$ for $E$ a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in $L_{2,\mathbb{Z}}$ given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just $\mathbb{Z}$.

17 pages. Since version 2 we extended the arguments from Z to more general rings

References in corpus (2)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Geometric classification of unital graph C*-algebras of real rank zero

Cited by in corpus (1)

• $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$