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.