Non-existence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

Abstract

Let n ≥ 2 , let R n be the graph consisting of one vertex and n loops and let R n − be its Cuntz splice. Let L n = L ( R n ) and L n − = L ( R n − ) be the Leavitt path algebras over a unital ring ℓ . Let C m be the cyclic group on 2 ≤ m ≤ ∞ elements. Equip L n and L n − with their natural C m -gradings. We show that under mild conditions on ℓ , which are satisfied, for example, when ℓ is a field or a principal ideal domain, there are no unital C m -graded ring homomorphisms L n → L n − nor in the opposite direction.