Actions of semigroups on directed graphs and their C^* -algebras

A free action α of a group G on a row-finite directed graph E induces an action α_* on its Cuntz-Krieger C^*-algebra C^*(E), and a recent theorem of Kumjian and Pask says that the crossed product C^*(E)×_α*G is stably isomorphic to the C^*-algebra C^*(E/G) of the quotient graph. We prove an analogue for free actions of Ore semigroups. The main ingredients are a new generalisation of a theorem of Gross and Tucker, dilation theory for endomorphic actions of Ore semigroups on graphs and C^*-algebras, and the Kumjian-Pask Theorem itself.