Crossed product Leavitt path algebras

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 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 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 G-group K-0(Gamma) (R).