Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions☆
60 citations recorded by the source corpus.
Abstract 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 L K ab ( E , C ) = C K ( Ω ( E , C ) ) ⋊ θ ⁎ alg F is 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 , 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.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/10d053d829e6d57b9b5acc70782cec7018d4ad3e