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

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 $\mathbb{Z}$.