The Cuntz splice does not preserve ⁎-isomorphism of Leavitt path algebras over Z

We show that the Leavitt path algebras L2,Z and L2−,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 LZ(E) for E a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in L2,Z given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just Z.