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

We attach to each finite bipartite separated graph (E,C)(E,C) a partial dynamical system (Ω(E,C),F,θ)(Ω(E,C),F,θ), where Ω(E,C)Ω(E,C) is a zero-dimensional metrizable compact space, FF is a finitely generated free group, and θ   is a continuous partial action of FF on Ω(E,C)Ω(E,C). The full crossed product C⁎-algebra O(E,C)=C(Ω(E,C))⋊θ⁎FO(E,C)=C(Ω(E,C))⋊θ⁎F is shown to be a canonical quotient of the graph C⁎-algebra C⁎(E,C)C⁎(E,C) of the separated graph (E,C)(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 LK(E,C)LK(E,C) of (E,C)(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 FF on a zero-dimensional metrizable compact space Z   such that the type semigroup S(Z,F,K)S(Z,F,K) is not almost unperforated, where KK denotes the algebra of clopen subsets of Z  . Finally we obtain a characterization of the separated graphs (E,C)(E,C) such that the canonical partial action of FF on Ω(E,C)Ω(E,C) is topologically free.