Nonexistence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

Abstract

Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.