Graded K-theory and Leavitt path algebras

Let G be a group and $$\ell $$ commutative unital $$*$$ -ring with an element $$\lambda \in \ell such that + \lambda ^*= 1$$ . We introduce variants of hermitian bivariant K-theory for -algebras equipped G-action or G-grading. For any graph E finitely many vertices weight function $$\omega :E^1 \rightarrow G$$ , distinguished triangle $$L(E)=L_\ell (E)$$ in the G-graded category $$kk^h_{G_{{\text {gr}}}}$$ is obtained, describing L(E) as cone matrix coefficients $$\mathbb {Z}[G]$$ associated to incidence In particular case standard {Z}$$ -grading, under mild assumptions on we show isomorphism class $$kk^h_{\mathbb {Z}_{{\text determined by graded Bowen–Franks module E. also obtain results general Leavitt path algebras which are independent interest, including versions Dade’s theorem Van den Bergh’s exact sequence relating ungraded K-theory.