Graded $K$-theory and Leavitt path algebras

Abstract

Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $\lambda \in \ell$ such that $\lambda + \lambda^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action or a $G$-grading. For any graph $E$ with finitely many vertices and any weight function $\omega \colon E^1 \to G$, a distinguished triangle for $L(E)=L_\ell(E)$ in the hermitian $G$-graded bivariant $K$-theory category $kk^h_{G_{\mathrm{gr}}}$ is obtained, describing $L(E)$ as a cone of a matrix with coefficients in $\mathbb{Z}[G]$ associated to the incidence matrix of $E$ and the weight $\omega$. In the particular case of the standard $\mathbb{Z}$-grading, and under mild assumptions on $\ell$, we show that the isomorphism class of $L(E)$ in $kk^h_{\mathbb{Z}_{\mathrm{gr}}}$ is determined by the graded Bowen-Franks module of $E$. We also obtain results for the graded and hermitian graded $K$-theory of $\ast$-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.