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Graphs, groupoids and CuntzKrieger algebras
, 1996
"... We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions the ..."
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We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C algebras are Morita equivalent; the groupoid C algebra C (G) is the CuntzKrieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C algebras using the general theory of Renault, and calculate their Ktheory. 1 Introduction Over the past fifteen years many C algebras and classes of C algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...
TOPOLOGICAL QUIVERS
, 2005
"... Abstract. Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C ∗correspondence, and from this correspondence one may construct a CuntzPimsner algebra C ∗ (Q). In thi ..."
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Abstract. Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C ∗correspondence, and from this correspondence one may construct a CuntzPimsner algebra C ∗ (Q). In this paper we develop the general theory of topological quiver C ∗algebras and show how certain C ∗algebras found in the literature may be viewed from this general perspective. In particular, we show that C ∗algebras of topological quivers generalize the wellstudied class of graph C ∗algebras and in analogy with that theory much of the operator algebra structure of C ∗ (Q) can be determined fromQ. We also show that many fundamental results from the theory of graph C ∗algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the GaugeInvariant Uniqueness theorem, the CuntzKrieger Uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity. 1.