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57
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 ...
On Groupoid C∗Algebras, Persistent Homology and TimeFrequency Analysis
"... We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in tim ..."
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Cited by 44 (1 self)
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We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in timefrequency analysis. The main result of our work is to illustrate how noncommutative C ∗algebras and the concept of Morita equivalence can be applied as a new type of analysis layer in signal processing. From a conceptual point of view, we use groupoid C∗algebras constructed with timefrequency data in order to study a given signal. From a computational point of view, we consider persistent homology as an algorithmic tool for estimating topological properties in timefrequency analysis. The usage of C∗algebras in our environment, together with the problem of designing computational algorithms, naturally leads to our proposal of using AFalgebras in the persistent homology setting. Finally, a computational toy example is presented, illustrating some elementary aspects of our framework. Due to the interdisciplinary nature
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 40 (11 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.
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 21 (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
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 18 (7 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
PROPERTY A, PARTIAL TRANSLATION STRUCTURES AND UNIFORM EMBEDDINGS IN GROUPS
, 2007
"... Abstract. We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C ∗algebra C ∗ (T) associated with it which is a subalgebra of the uniform Roe algebra C ∗ u(X). We introduce a coarse invariant of the metric which provides an obstruction t ..."
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Cited by 17 (2 self)
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Abstract. We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C ∗algebra C ∗ (T) associated with it which is a subalgebra of the uniform Roe algebra C ∗ u(X). We introduce a coarse invariant of the metric which provides an obstruction to embedding the space in a group. When the space is sufficiently grouplike, as determined by our invariant, properties of the Roe algebra can be deduced from those of C ∗ (T). We also give a proof of the fact that the uniform Roe algebra of a metric space is a coarse invariant up to Morita equivalence. Many interesting geometric properties of spaces and groups are captured by the structure of C ∗algebras associated with those objects. For example, a discrete group G is amenable if and only if the full C ∗algebra C ∗ (G) is nuclear [7]. In a similar vein, for a discrete group G, Yu’s property A is equivalent both to the nuclearity of the uniform Roe algebra C ∗ u (G) and to the exactness of the reduced C∗algebra C ∗ r (G). This follows from the results of AnantharamanDelaroche and Renault [1], Higson and Roe [5], Guentner and Kaminker [4], and Ozawa [10]. While property A and the uniform Roe algebra can be defined for arbitrary metric spaces, we cannot generalise these results without a good analogue of the reduced C ∗algebra of a group. In this paper we introduce a C∗algebra to fulfill this role. To do so we carry out the following programme. First we define the notion of a partial translation structure (Definition 11) on a uniformly discrete metric space, which captures geometrically the interplay between the left and the right action of a group on itself. In broad terms, this can be described as follows. In Euclidean space translations
The structure of C*algebras associated with hyperbolic dynamical systems
 J. Funct. Anal
, 1999
"... The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of sresolving, finitetoone fa ..."
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Cited by 15 (3 self)
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The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of sresolving, finitetoone factor maps, then the construction of the stable C*algebra is contravariant, while that of the unstable C*algebra is covariant. The paper also discusses the constructions of these C*algebras for Smale spaces which are not mixing. 1. Introduction and
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 14 (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.
STRONG MORITA EQUIVALENCE OF INVERSE SEMIGROUPS
, 2009
"... We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C ..."
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Cited by 13 (4 self)
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We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C ∗algebras. As a consequence we obtain a new proof of a result of Khoshkam and Skandalis showing that the C ∗algebra of an Finverse semigroup is strongly Morita equivalent to a cross product of a commutative C ∗algebra by a group.