Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

TL;DR

This paper proves that certain module maps between graded Grothendieck groups of Leavitt path algebras can be lifted to algebra homomorphisms, confirming part of Hazrat's conjecture for fields.

Contribution

It introduces a combinatorial method to lift module maps to algebra homomorphisms and verifies the fullness of Hazrat's functor for fields.

Findings

Module maps lift to algebra homomorphisms for finite graphs.

Established the fullness of Hazrat's functor over fields.

Characterized maps as scalar extensions preserving a sub-$ ext{ast}$-semiring.

Abstract

We show that any pointed, preordered module map $\mathfrak{BF}_{\mathrm{gr}}(E) \to \mathfrak{BF}_{\mathrm{gr}}(F)$ between Bowen-Franks modules of finite graphs can be lifted to a unital, graded, diagonal preserving $\ast$-homomorphism $L_\ell(E) \to L_\ell(F)$ between the corresponding Leavitt path algebras over any commutative unital ring with involution $\ell$. Specializing to the case when $\ell$ is a field, we establish the fullness part of Hazrat's conjecture about the functor from Leavitt path $\ell$-algebras of finite graphs to preordered modules with order unit that maps $L_\ell(E)$ 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 $\ell$ of unital, graded $\ast$-homomorphisms $L_{\mathbb Z}(E) \to L_{\mathbb Z}(F)$ that preserve a sub-$\ast$-semiring introduced here.