Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

Abstract

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 Lℓ(E)→Lℓ(F) between the corresponding Leavitt path algebras over any commutative unital ring with involution ℓ. Specializing to the case when ℓ is a field, we establish the fullness part of Hazrat's conjecture about the functor from Leavitt path ℓ-algebras of finite graphs to preordered modules with order unit that maps Lℓ(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 ℓ of unital, graded ⁎-homomorphisms LZ(E)→LZ(F) that preserve a sub-⁎-semiring introduced here.