Crossed product Leavitt path algebras

If E is a directed graph and K is a field, the Leavitt path algebra LK(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 LK(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 LK(E) are also characterized in terms of the pre-ordered group properties of the Grothendieck ℤ-group of LK(E). If E has finitely many vertices, we characterize when LK(E) is strongly graded in terms of the properties of K0Γ(L K(E)). Our proof also provides an alternative to the known proof of the equivalence LK(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 LK(E) is strongly graded and graded unit-regular if and only if LK(E) is a crossed product. In the process of showing the main result, we obtain conditions on a group Γ and a Γ-graded division ring K equivalent with the requirements that a Γ-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 Γ on the Grothendieck Γ-group K0Γ(R).