Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

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.

19 pages. Version accepted for publication. Corrected the statement of Theorem 6.16 (now Theorem 6.17)

Cited by in corpus (2)

• Classification conjectures for Leavitt path algebras
• Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras