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34
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 ...
The equivariant Brauer group of a locally compact groupoid
, 1997
"... 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 ..."
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Cited by 39 (16 self)
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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.
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 23 (5 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 16 (6 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.
The caloron correspondence and higher string classes for loop groups
 J. Geom. Phys
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Cocycles over partially hyperbolic maps
 Math. USSR Izvestija 8 (1974), 177–218. COHOMOLOGICAL EQUATION 93
, 1945
"... Abstract. We give a general necessary condition for the extremal (largest and smallest) Lyapunov exponents of a Hölder continuous cocycle over a volume preserving partially hyperbolic diffeomorphism to coincide. This condition applies to smooth cocycles, with linear and projective cocycles as speci ..."
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Cited by 9 (3 self)
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Abstract. We give a general necessary condition for the extremal (largest and smallest) Lyapunov exponents of a Hölder continuous cocycle over a volume preserving partially hyperbolic diffeomorphism to coincide. This condition applies to smooth cocycles, with linear and projective cocycles as special cases. It is based on an abstract rigidity result for fiber bundle sections that are holonomyinvariant, or even just continuous, over the strongstable leaves and the strongunstable leaves of the diffeomorphism. As an application, we prove that the subset of Hölder continuous linear cocycles for which the extremal Lyapunov exponents do coincide is meager and even has infinite codimension. Contents
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 8 (2 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
Isotropy of semisimple group actions on manifolds with geometric structure
 Amer. J. Math
, 1998
"... of semisimple group actions on manifolds with geometric structure ..."
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Cited by 8 (0 self)
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of semisimple group actions on manifolds with geometric structure
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 7 (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.
Classi of simple Calgebras with unique traces
 Amer. J. Math
, 1998
"... Abstract. We classify certain almost multiplicative morphisms up to approximate unitary equivalence and use this result to prove the following: Let A and B be two unital separable simple Calgebras of real rank zero, stable rank one, with weakly unperforated K0groups and with unique normalized qua ..."
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Cited by 4 (2 self)
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Abstract. We classify certain almost multiplicative morphisms up to approximate unitary equivalence and use this result to prove the following: Let A and B be two unital separable simple Calgebras of real rank zero, stable rank one, with weakly unperforated K0groups and with unique normalized quasitraces. Suppose that both A and B are locally AH and (K (A), K