Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions

We attach to each finite bipartite separated graph (E, C) a partial dynamical system (Ω(E,C),F,θ), where Ω(E, C) is a zero-dimensional metrizable compact space, F is a finitely generated free group, and θ is a continuous partial action of F on Ω(E, C). The full crossed product C*-algebra O(E,C)=C(Ω(E,C))⋊θ*F is shown to be a canonical quotient of the graph C*-algebra C *(E, C) of the separated graph (E, C). Similarly, we prove that, for any *-field K, the algebraic crossed product LKab(E,C)=CK(Ω(E,C))⋊θ*algF is a canonical quotient of the Leavitt path algebra L K(E, C) of (E, C). The monoid V(LKab(E,C)) of isomorphism classes of finitely generated projective modules over LKab(E,C) is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs. Using this, we are able to construct an action of a finitely generated free group F on a zero-dimensional metrizable compact space Z such that the type semigroup S(Z,F,K) is not almost unperforated, where K denotes the algebra of clopen subsets of Z. Finally we obtain a characterization of the separated graphs (E, C) such that the canonical partial action of F on Ω(E, C) is topologically free. © 2013 Elsevier Inc.