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.

7 pages. Second version fixes the statement of old Proposition 6.4, now Proposition 6.3