Actions of semigroups on directed graphs and their [formula omitted]-algebras

Abstract

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.