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46
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 25 (11 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 14 (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
Quantized reduction as a tensor product
 QUANTIZATION OF SINGULAR SYMPLECTIC QUOTIENTS. BASEL: BIRKHÄUSER, 2001. EPRINT MATHPH/0008004
, 2008
"... Matched bimodules for rings may be composed through the (algebraic) bimodule tensor product, the canonical bimodule R → R ← R serving as a unit for ⊗R. We describe this picture also for C ∗algebras, von Neumann algebras, Lie groupoids, Poisson manifolds, and symplectic groupoids. This hinges on th ..."
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Cited by 13 (5 self)
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Matched bimodules for rings may be composed through the (algebraic) bimodule tensor product, the canonical bimodule R → R ← R serving as a unit for ⊗R. We describe this picture also for C ∗algebras, von Neumann algebras, Lie groupoids, Poisson manifolds, and symplectic groupoids. This hinges on the correct notion of bimodule, tensor product, and unit: for C ∗algebras B one has Hilbert (C ∗ ) bimodules with Rieffel’s tensor product and the canonical Hilbert bimodule over B, for von Neumann algebras one uses correspondences with Connes’s tensor product and the standard form, for (symplectic) Lie groupoids G one has regular (symplectic) bibundles with the Hilsum–Skandalis tensor product and the canonical bibundle over G, and for integrable Poisson manifolds P one deals with regular symplectic bimodules (dual pairs) with Xu’s tensor product and the sconnected and ssimply connected symplectic groupoid over P. Morita theory relates socalled equivalence bimodules to equivalence of representation theories. Subsequently, we study certain interconnections between the various constructions. The relation between Hilbert bimodules and correspondences is reviewed in detail. The notion of Marsden–Weinstein reduction makes sense for Poisson manifolds, C ∗algebras, and von Neumann algebras. Poisson manifolds and Lie groupoids join in the theory of Lie algebroids and symplectic groupoids. Finally, we note that the Poisson manifolds associated to Morita equivalent sconnected and ssimply connected Lie groupoids are Morita equivalent in the sense of Xu.
C*Algebras of Directed Graphs and Group Actions
, 1997
"... Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G ..."
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Cited by 11 (4 self)
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Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G we may use our results to show that C*(E) is strongly Morita equivalent to the crossed product C0 (@T ) \Theta G, where @T is a certain 0dimensional space canonically associated to the tree.
Arithmetic Manifolds of Positive Scalar Curvature
 J. Differential Geom
, 1999
"... this paper is to explore the situation if we study the problem of complete metrics with no quasiisometry conditions at all. ..."
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Cited by 9 (3 self)
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this paper is to explore the situation if we study the problem of complete metrics with no quasiisometry conditions at all.
Continuous family groupoids
 Homology, Homotopy and Applications
"... Abstract. In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant AtiyahSinger index theorem for families, and is also required in noncommutative geometry. T ..."
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Cited by 9 (4 self)
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Abstract. In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant AtiyahSinger index theorem for families, and is also required in noncommutative geometry. The class includes that of Lie groupoids, and the paper shows that, like Lie groupoids, continuous family groupoids always admit (an essentially unique) continuous left Haar system of smooth measures. We also show that the action of a continuous family groupoid G on a continuous family Gspace fibered over another continuous family Gspace Y can always be regarded as an action of the continuous family groupoid G ∗ Y on an ordinary G ∗ Yspace. 1.
GROUPOIDS AND AN INDEX THEOREM FOR CONICAL PSEUDOMANIFOLDS
, 2006
"... Abstract. We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M. A main ingredient is a noncommut ..."
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Cited by 8 (4 self)
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Abstract. We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M. A main ingredient is a noncommutative algebra that plays in our setting the role of C0(T ∗ M). We prove a Thom isomorphism between noncommutative algebras which gives a new example of wrong way functoriality in Ktheory. We then give a new proof of the AtiyahSinger index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds. Contents
The Fourier algebra for locally compact groupoids
, 2003
"... We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compac ..."
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Cited by 8 (2 self)
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We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
A Path Model for Circle Algebras
 JOT
, 1995
"... Using groupoid theory, we construct a path model for finite type embeddings of circle algebras that generalizes the path model of Ocneanu and Sunder for Bratteli diagrams. The JonesWatatani index is computed using the maps induced on K0theory by the embedding and its dual. The analysis is based on ..."
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Cited by 8 (4 self)
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Using groupoid theory, we construct a path model for finite type embeddings of circle algebras that generalizes the path model of Ocneanu and Sunder for Bratteli diagrams. The JonesWatatani index is computed using the maps induced on K0theory by the embedding and its dual. The analysis is based on imprimitivity groupoids associated to the embeddings. Taking inductive limits, we obtain generalizations of the BunceDeddens algebras. 1991 Mathematical Subject Classification: Primary 46L05, 46L55. The notion of Bratteli diagram was introduced in [Br1], and since then, it has played an important rôle in the study of AFalgebras. Subsequently, Ocneanu and Sunder introduced a path model for inclusions of finite dimensional algebras and used it to analyze the index theory for subfactors. In this paper, we investigate algebras that arise
Operator Algebras and Poisson Manifolds Associated to Groupoids
, 2001
"... It is well known that a measured groupoid G defines a von Neumann algebra W # (G), and that a Lie groupoid G canonically defines both a C # algebra C # (G) and a Poisson manifold A # (G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C # algebras, and ..."
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Cited by 7 (3 self)
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It is well known that a measured groupoid G defines a von Neumann algebra W # (G), and that a Lie groupoid G canonically defines both a C # algebra C # (G) and a Poisson manifold A # (G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C # algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G C # (G), and G are functorial between the categories in question. It follows that these maps preserve Morita equivalence.