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 (Omega (E, C), F, theta), where Omega(E, C) is a zero-dimensional metrizable compact space, F is a finitely generated free group, and theta is a continuous partial action of F on Omega(E, C). The full crossed product C*-algebra O(E, C) = C (Omega(E, C)) theta* 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 L-K(ab)(E, C) = C-K(Omega(E, C)) (alg)(theta*) IF s a canonical quotient of the Leavitt path algebra L-K(E, C) of (E, C). The monoid V(L-K(ab)(E, C)) of isomorphism classes of finitely generated projective modules over L-K(ab)(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, IF, 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 Omega(E, C) is topologically free. (C) 2013 Elsevier Inc. All rights reserved.