Graded $K$-theory and Leavitt path algebras

Abstract

Let G be a group and ℓ a commutative unital * -ring with an element λ ∈ ℓ such that λ + λ * = 1. We introduce variants of hermitian bivariant K-theory for * -algebras equipped with a G-action or a G-grading. For any graph E with finitely many vertices and any weight function ω :Ggr is obtained, describing L(E) as a cone of a matrix with coefficients in Z[G] associated to the incidence matrix of E and the weight ω. In the particular case of the standard Z-grading, and under mild assumptions on ℓ, we show that the isomorphism class of L(E) in kk hZgr is determined by the graded Bowen-Franks module of E. We also obtain results for the graded and hermitian graded K-theory of * -algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade's theorem and of Van den Bergh's exact sequence relating graded and ungraded K-theory.Again the same formulas hold for homotopy algebraic and homotopy hermitian K-theory without any unitality assumptions on R. We also consider gradings on L(E) with values on a group G for an arbitrary graph E.

Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

“…. By [4,Corollary 5.4] the map above is an isomorphism whenever the Grothendieck group of ℓ is isomorphic to Z; for example, such is the case when ℓ is a field or more generally a PID. Remark 2.4.2.…”

“…If R is a ring with local units, then by [2, Section 4A] the monoid V ∞ (R) is isomorphic to the monoid of isomorphism classes of finitely generated projective unital R-modules. Let S be a unital Z-graded ring and let Z ⋉ S be its crossed product as defined in [4,Subsection 2.5]. By [2, Section 2C] (see also [4,Section 3.1]), we have that V ∞ (Z ⋉ S) is naturally isomorphic to the monoid V gr (S) of isomorphism classes of finitely generated Z-graded projective S-modules.…”

“…Let S be a unital Z-graded ring and let Z ⋉ S be its crossed product as defined in [4,Subsection 2.5]. By [2, Section 2C] (see also [4,Section 3.1]), we have that V ∞ (Z ⋉ S) is naturally isomorphic to the monoid V gr (S) of isomorphism classes of finitely generated Z-graded projective S-modules. In particular K gr 0 (S) is the group completion of V ∞ (Z ⋉ S).…”

“…Let E be a finite graph. We first recall from [4] a characterization of BF gr (E) and establish further properties of this presentation. For each k ∈ Z and v ∈ E 0 , write…”

“…Section 3 is dedicated to auxiliary results regarding the relation between the graded Grothendieck group and homogeneous idempotents. In Section 4 we establish some properties of Bowen-Franks modules using a presentation introduced in [4]; this is then used in Section 5 to give a characterization of maps between Bowen-Franks modules in terms of vertex indexed matrices with nonnegative integer coefficients (Proposition 5.1). With this in place, in Section 6 we prove Theorem 1.4 as Theorem 6.1; this section also contains the proof of Theorem 1.5 as Theorem 6.16.…”