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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 ...
Generalized CuntzKrieger algebras
 Proc. AMS
"... To a special embedding Φ of circle algebras having the same spectrum, we associate an rdiscrete, locally compact groupoid, similar to the CuntzKrieger groupoid. Its C∗algebra, denoted OΦ, is a continuous version of the CuntzKrieger algebras OA. The algebra OΦ is generated by an ATalgebra and a ..."
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Cited by 20 (6 self)
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To a special embedding Φ of circle algebras having the same spectrum, we associate an rdiscrete, locally compact groupoid, similar to the CuntzKrieger groupoid. Its C∗algebra, denoted OΦ, is a continuous version of the CuntzKrieger algebras OA. The algebra OΦ is generated by an ATalgebra and a nonunitary isometry. We compute its Ktheory under the assumption that the ATalgebra is simple. 1991 Mathematical Subject Classification: Primary 46L05, 46L55, 46L80. The idea of considering continuous versions of the CuntzKrieger algebras OA occured to us as a result of studying the groupoid approach to the Cuntz algebra On. In [De1], we treated continuous versions of On, replacing the Cantor set and the unilateral shift by a compact space and a selfcovering. In this paper we are concerned with a local homeomorphism σ on
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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Cited by 1 (0 self)
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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.
C∗ALGEBRAS ASSOCIATED WITH TOPOLOGICAL GROUP QUIVERS I: GENERATORS, RELATIONS AND SPATIAL STRUCTURE
"... Abstract. Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver Q is a C∗correspondence, and in turn, a CuntzPimsner algebra C∗(Q). Given Γ a locally compact g ..."
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Cited by 1 (1 self)
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Abstract. Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver Q is a C∗correspondence, and in turn, a CuntzPimsner algebra C∗(Q). Given Γ a locally compact group and α and β endomorphisms on Γ, one may construct a topological quiver Qα,β(Γ) with vertex set Γ, and edge set Ωα,β(Γ) = {(x, y) ∈ Γ × Γ ∣∣α(y) = β(x)}. In this paper, the author examines the CuntzPimsner algebra Oα,β(Γ): = C∗(Qα,β(Γ)). The investigative topics include a notion for topological quiver isomorphisms, generators (and their relations) of the C∗algebras Oα,β(Γ), and its spatial structure (i.e., colimits, tensor products and crossed products) and a few properties of its C∗subalgebras.
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