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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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Cited by 48 (18 self)
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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 ...
C*ALGEBRAS ASSOCIATED WITH INTERVAL MAPS
"... Abstract. For each piecewise monotonic map τ of [0, 1], we associate a pair of C*algebras Fτ and Oτ and calculate their Kgroups. The algebra Fτ is an AIalgebra. We characterize when Fτ and Oτ are simple. In those cases, Fτ has a unique trace, and Oτ is purely infinite with a unique KMSstate. In ..."
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Cited by 10 (2 self)
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Abstract. For each piecewise monotonic map τ of [0, 1], we associate a pair of C*algebras Fτ and Oτ and calculate their Kgroups. The algebra Fτ is an AIalgebra. We characterize when Fτ and Oτ are simple. In those cases, Fτ has a unique trace, and Oτ is purely infinite with a unique KMSstate. In the case that τ is Markov, these algebras include the CuntzKrieger algebras OA, and the associated AFalgebras FA. Other examples for which the Kgroups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and βtransformations. For the case of interval exchange maps and of βtransformations, the C*algebra Oτ coincides with the algebras defined by Putnam and KatayamaMatsumotoWatatani respectively. 1.
AND
, 1995
"... 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 one–sided 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 ..."
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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 one–sided 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 Cuntz–Krieger algebra of an infinite {0, 1}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