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. The space of paths in a directed graph E can be modelled by systems of partial isometries on Hilbert space: one associates to each edge e a partial isometry Se in such a way that the product SeSf is a nonzero partial isometry when ef is a path, and zero otherwise. Since partial isometries are the linear operators T satisfying T = TT ∗T , the appropriate algebraic envelopes for such Cuntz-Krieger systems {Se} are C∗-algebras; the graph algebra C∗(E) is the universal C∗-algebra generated by a Cuntz-Krieger system. For finite graphs, these algebras turn out to be precisely the Cuntz-Krieger algebras associated to Markov chains [3]. More recently, the algebras of infinite graphs have arisen in a variety of contexts, and the fundamental results of Cuntz and Krieger extend in an attractive and approachable manner (see [2] for more precise statements and references). The general theory of C∗-algebras provides powerful tools for problems involving symmetry groups and representation theory, so it is natural to ask how group actions interact with graph algebras. Every action α of a group G on E induces an action α∗ of G on C∗(E). A recent theorem of Kumjian and Pask asserts that if α is free, then the crossed product C(E)×α∗ G is stably isomorphic to the C∗-algebra C∗(E/G) of the quotient graph [10, Corollary 3.10]; alternatively, the C∗-algebras C∗(E) ×α∗ G and C∗(E/G) are Morita equivalent in the sense appropriate for C∗-algebras. This theorem is strikingly similar to a well-known result of Green about free and proper actions of groups on locally compact spaces [5], and suggests that graph algebras might be profitably viewed as noncommutative function algebras in the sense of Connes. Here we prove an analogue of the Kumjian-Pask Theorem for endomorphic actions of semigroups on graphs, under mild hypotheses on the semigroup and the action. First, we restrict our attention to Ore semigroups S, which can be always be embedded in a group Γ in such a way that Γ = S−1S. We do this because Laca has shown that one can then expect to dilate endomorphic actions of S to automorphic actions of Γ [12]. The class of Ore semigroups includes all generating subsemigroups of abelian groups as well as many interesting nonabelian semigroups (see [12, §1.1], for example). Second, we assume that the action of S has a fundamental domain: a collection of edges and their sources whose images under S cover E. There is always such a domain for group actions, but simple examples show that we need to assume it Date: 8 November 1999. 1991 Mathematics Subject Classification. 20M20, 46L55, 54H15, 05C20.