Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

We show that any pointed, preordered module map BFgr(E)-+ BFgr(F) between Bowen-Franks modules of finite graphs can be lifted to a unital, graded, diagonal preserving *homomorphism Lt(E) -+ Lt(F) between the corresponding Leavitt path algebras over any commutative unital ring with involution $. Specializing to the case when $ is a field, we establish the fullness part of Hazrat's conjecture about the functor from Leavitt path $-algebras of finite graphs to preordered modules with order unit that maps Lt(E) to its graded Grothendieck group. Our construction of lifts is of combinatorial nature; we characterize the maps arising from this construction as the scalar extensions along $ of unital, graded *-homomorphisms LZ(E) -+ LZ(F) that preserve a sub-*-semiring introduced here.& COPY; 2023 Elsevier Inc. All rights reserved.