Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

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