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)) \rtimes_{\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^{ab}_K(E,C) = C_K(\Omega(E,C)) \rtimes_\theta^{alg} F is a canonical quotient of the Leavitt path algebra L_K(E,C) of (E,C). The monoid V(L^{ab}_K(E,C)) of isomorphism classes of finitely generated projective modules over L^{ab}_K(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 \Omega(E,C) is topologically free.