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

Abstract

Let [Formula: see text], let [Formula: see text] be the graph consisting of one vertex and [Formula: see text] loops and let [Formula: see text] be its Cuntz splice. Let [Formula: see text] and [Formula: see text] be the Leavitt path algebras over a unital ring [Formula: see text]. Let [Formula: see text] be the cyclic group on [Formula: see text] elements. Equip [Formula: see text] and [Formula: see text] with their natural [Formula: see text]-gradings. We show that under mild conditions on [Formula: see text], which are satisfied, for example, when [Formula: see text] is a field or a principal ideal domain, there are no unital [Formula: see text]-graded ring homomorphisms [Formula: see text] nor in the opposite direction.