234 research outputs found

    Simplicity of 2-graph algebras associated to Dynamical Systems

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    We give a combinatorial description of a family of 2-graphs which subsumes those described by Pask, Raeburn and Weaver. Each 2-graph Ξ›\Lambda we consider has an associated Cβˆ—C^*-algebra, denoted Cβˆ—(Ξ›)C^*(\Lambda), which is simple and purely infinite when Ξ›\Lambda is aperiodic. We give new, straightforward conditions which ensure that Ξ›\Lambda is aperiodic. These conditions are highly tractable as we only need to consider the finite set of vertices of Ξ›\Lambda in order to identify aperiodicity. In addition, the path space of each 2-graph can be realised as a two-dimensional dynamical system which we show must have zero entropy.Comment: 19 page

    The Noncommutative Geometry of Graph Cβˆ—C^*-Algebras I: The Index Theorem

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    We investigate conditions on a graph Cβˆ—C^*-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth (1,∞)(1,\infty)-summable semfinite spectral triple. The local index theorem allows us to compute the pairing with KK-theory. This produces invariants in the KK-theory of the fixed point algebra, and these are invariants for a finer structure than the isomorphism class of Cβˆ—(E)C^*(E).Comment: 33 page

    A dual graph construction for higher-rank graphs, and KK-theory for finite 2-graphs

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    Given a kk-graph Ξ›\Lambda and an element pp of \NN^k, we define the dual kk-graph, pΞ›p\Lambda. We show that when Ξ›\Lambda is row-finite and has no sources, the Cβˆ—C^*-algebras Cβˆ—(Ξ›)C^*(\Lambda) and Cβˆ—(pΞ›)C^*(p\Lambda) coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the KK-theory of Cβˆ—(Ξ›)C^*(\Lambda) when Ξ›\Lambda is finite and strongly connected and satisfies the aperiodicity condition.Comment: 9 page

    Group actions on labeled graphs and their C*-algebras

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    We introduce the notion of the action of a group on a labeled graph and the quotient object, also a labeled graph. We define a skew product labeled graph and use it to prove a version of the Gross-Tucker theorem for labeled graphs. We then apply these results to the Cβˆ—C^*-algebra associated to a labeled graph and provide some applications in nonabelian duality.Comment: 18 pages, updated versio
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