Crossed product Leavitt path algebras

Abstract

If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb Z.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of $L_K(E)$ are also characterized in terms of the pre-ordered group properties of the Grothendieck $\mathbb Z$-group of $L_K(E)$. If $E$ has finitely many vertices, we characterize when $L_K(E)$ is strongly graded in terms of the properties of $K_0^\Gamma(L_K(E)).$ Our proof also provides an alternative to the known proof of the equivalence $L_K(E)$ is strongly graded if and only if $E$ has no sinks for a finite graph $E.$ We also show that, if unital, the algebra $L_K(E)$ is strongly graded and graded unit-regular if and only if $L_K(E)$ is a crossed product. In the process of showing the main result, we obtain conditions on a group $\Gamma$ and a $\Gamma$-graded division ring $K$ equivalent with the requirements that a $\Gamma$-graded matrix ring $R$ over $K$ is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group $\Gamma$ on the Grothendieck $\Gamma$-group $K_0^\Gamma(R).$