  We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and
$L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-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
$\mathbb{Z}$.
