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14
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 ...
The completion theorem in Ktheory for proper actions of a discrete group
"... We prove a version of the AtiyahSegal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper GCWcomplex X, K (EG\Theta GX) is the completion of K G (X) with respect to a certain ideal. We also show, for such G and X, that KG (X) can be ..."
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Cited by 21 (5 self)
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We prove a version of the AtiyahSegal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper GCWcomplex X, K (EG\Theta GX) is the completion of K G (X) with respect to a certain ideal. We also show, for such G and X, that KG (X) can be defined as the Grothendieck group of the monoid of Gvector bundles over X. Key words: Ktheory, proper actions, vector bundles 1991 mathematics subject classification: primary 55N91, secondary 19L47 Let G be any discrete group. For such G, a GCWcomplex is a CWcomplex with G action which permutes the cells, such that an element g 2 G sends a cell to itself only by the identity map. A GCWcomplex X is proper if all of its isotropy subgroups have finite order, and is finite if it is made up of finitely many orbits of cells. A GCWpair is a pair of Gspaces (X; A), where X is a GCWcomplex and A is a Ginvariant subcomplex. The main results of this paper are Theorems 3.2 and 4.3 below...
The equivariant Brauer group of a locally compact groupoid
, 1996
"... Abstract. We define the Brauer group Br(G) of a locally compact groupoid G to be the set of Morita equivalence classes of pairs (A, α) consisting of an elementary C ∗bundle A over G (0) satisfying Fell’s condition and an action α of G on A by ∗isomorphisms. When G is the transformation groupoid X ..."
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Cited by 17 (8 self)
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Abstract. We define the Brauer group Br(G) of a locally compact groupoid G to be the set of Morita equivalence classes of pairs (A, α) consisting of an elementary C ∗bundle A over G (0) satisfying Fell’s condition and an action α of G on A by ∗isomorphisms. When G is the transformation groupoid X × H, then Br(G) is the equivariant Brauer group BrH(X). In addition to proving that Br(G) is a group, we prove three isomorphism results. First we show that if G and H are equivalent groupoids, then Br(G) and Br(H) are isomorphic. This generalizes the result that if G and H are groups acting freely and properly on a space X, say G on the left and H on the right then BrG(X/H) and BrH(G\X) are isomorphic. Secondly we show that the subgroup Br0(G) of Br(G) consisting of classes [A, α] with A having trivial DixmierDouady invariant is isomorphic to a quotient E(G) of the collection Tw(G) of twists over G. Finally we prove that Br(G) is isomorphic to the inductive limit Ext(G, T) of the groups E(G X) where X varies over all principal G spaces X and G X is the imprimitivity groupoid associated to X. 1.
Triviality of Bloch and BlochDirac bundles
 Ann. Henri Poincaré
, 2007
"... In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasiBloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the timereversal symmetry of the Hamilton ..."
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Cited by 8 (0 self)
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In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasiBloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the timereversal symmetry of the Hamiltonian and some bundletheoretic methods, we show that the problem has a positive answer in any dimension d ≤ 3, thus generalizing a previous result by G. Nenciu. We provide a general formulation of the result, aiming at the application to the Dirac equation with a periodic potential and to piezoelectricity. 1
Crossed products by C0(X)actions
 J. Funct. Anal
, 1998
"... Dedicated to Professor E. Kaniuth on the occasion of his 60 th birthday Abstract. Suppose that G has a representation group H, that Gab: = G/[G, G] is compactly generated, and that A is a C ∗algebra for which the complete regularization of Prim(A) is a locally compact Hausdorff space X. In a previo ..."
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Cited by 7 (4 self)
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Dedicated to Professor E. Kaniuth on the occasion of his 60 th birthday Abstract. Suppose that G has a representation group H, that Gab: = G/[G, G] is compactly generated, and that A is a C ∗algebra for which the complete regularization of Prim(A) is a locally compact Hausdorff space X. In a previous article, we showed that there is a bijection α ↦ → (Zα, fα) between the collection of exterior equivalence classes of locally inner actions α: G → Aut(A), and the collection of principal ̂ Gabbundles Zα together with continuous functions fα: X → H 2 (G, T). In this paper, we compute the crossed products A ⋊α G in terms of the data Zα, fα, and C ∗ (H). 1.
A Homotopy Theory of Orbispaces
"... An orbifold is a singular space which is locally modeled on the quotient of a smooth manifold by a smooth action of a finite group. It appears naturally in geometry and topology when group actions on manifolds are involved and the stabilizer of each fixed point is finite. The concept of an orbifold ..."
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Cited by 6 (1 self)
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An orbifold is a singular space which is locally modeled on the quotient of a smooth manifold by a smooth action of a finite group. It appears naturally in geometry and topology when group actions on manifolds are involved and the stabilizer of each fixed point is finite. The concept of an orbifold was first introduced by Satake under the name “Vmanifold ” in a paper where he also extended the basic differential geometry to his newly defined singular spaces (cf. [Sa]). The local structure of an orbifold – being the quotient of a smooth manifold by a finite group action – was merely used as some “generalized smooth structure”. A different aspect of the local structure was later recognized by Thurston, who gave the name “orbifold ” and introduced an important concept – the fundamental group of an orbifold (cf. [Th]). In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on CalabiYau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist’s Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related to the so called McKay correspondence in mathematics (cf. [McK]). Later developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of
On the Cohomology Algebra of Free Loop Spaces
"... Let X be a simply connected space and K be any field. The normalized singular cochains N (X ; K) admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology HH N X of the space X . We prove that, endowed with this product, HH N ..."
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Cited by 5 (0 self)
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Let X be a simply connected space and K be any field. The normalized singular cochains N (X ; K) admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology HH N X of the space X . We prove that, endowed with this product, HH N X is isomorphic to the cohomology algebra of the free loop space of X with coefficients in K. We also show how to construct a simpler Hochschild complex which allows direct computation.
Are Hamiltonian Flows Geodesic Flows?
, 2002
"... When a Hamiltonian system has a "Kinetic + Potential" structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure, so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic fl ..."
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Cited by 2 (1 self)
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When a Hamiltonian system has a "Kinetic + Potential" structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure, so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.
The DixmierDouady invariant for Dummies
 Notices Amer. Math. Soc
, 2009
"... The DixmierDouady invariant is the primary tool in the classification of continuous trace C∗algebras. These algebras have come to the fore in recent years because of their relationship to twisted Ktheory and via twisted Ktheory to branes, gerbes, and string theory. This note sets forth the basic ..."
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The DixmierDouady invariant is the primary tool in the classification of continuous trace C∗algebras. These algebras have come to the fore in recent years because of their relationship to twisted Ktheory and via twisted Ktheory to branes, gerbes, and string theory. This note sets forth the basic properties of the DixmierDouady invariant using only classical homotopy and bundle theory. Algebraic topology enters the scene at once since the algebras in question are algebras of sections of certain fibre bundles. The results stated are all contained in the original papers of Dixmier and Douady [5], Donovan and Karoubi [7], and Rosenberg [23]. Our treatment is novel in that it avoids the sheaftheoretic techniques of the original proofs and substitutes more classical algebraic topology. Some of the proofs are borrowed directly from the recent paper of Atiyah and Segal [1]. Those interested in more detail and especially in the connections with analysis should consult Rosenberg [23], the definitive work of Raeburn and Williams [21], as well as the recent paper of Karoubi [13] and the book by Cuntz, Meyer, and Rosenberg [4]. We briefly discuss twisted Ktheory itself, mostly in order to direct the interested reader to some of the (exponentiallygrowing) literature on the subject. It is a pleasure to acknowledge the assistance of Alan Carey, Dan Isaksen, Max Karoubi, N. C. Phillips, and Jonathan Rosenberg in the preparation of this paper. Fibre Bundles Suppose that G is a topological group and G → T → X is a principal Gbundle over the compact space X. Then up to equivalence it is classified by Claude Schochet is professor of mathematics at Wayne State University, Detroit, MI. His email address is claude@ math.wayne.edu. a map f to the classifying space BG and there is a pullback diagram