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 $\mathbb{ Z}$.
Categories: math.RA math.OA