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15
Noncommutative correspondences, duality and Dbranes in bivariant Ktheory
, 2007
"... We describe a categorical framework for the classification of Dbranes on noncommutative spaces using techniques from bivariant Ktheory of C∗algebras. We present a new description of bivariant Ktheory in terms of noncommutative correspondences which is nicely adapted to the study of Tduality in ..."
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We describe a categorical framework for the classification of Dbranes on noncommutative spaces using techniques from bivariant Ktheory of C∗algebras. We present a new description of bivariant Ktheory in terms of noncommutative correspondences which is nicely adapted to the study of Tduality in open string theory. We systematically use the diagram calculus for bivariant Ktheory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.
Equivalences of smooth and continuous principal bundles with infinitedimensional structure groups
, 2008
"... Let K be a a Lie group, modeled on a locally convex space, and M a finitedimensional paracompact manifold with corners. We show that each continuous principal Kbundle over M is continuously equivalent to a smooth one and that two smooth principal Kbundles over M which are continuously equivalent ..."
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Let K be a a Lie group, modeled on a locally convex space, and M a finitedimensional paracompact manifold with corners. We show that each continuous principal Kbundle over M is continuously equivalent to a smooth one and that two smooth principal Kbundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.
Projective Dirac operators, twisted Ktheory and local index formula, arXiv:1008.0707
 J. Noncommutative Geom. Mathematics Department, Caltech, 1200 E. California Blvd. Pasadena, CA 91125, USA Email
"... I am very grateful to BaiLing Wang. He foresaw the possibility that the projective spin Dirac operator defined by [19] in formal sense can be realized by a certain spectral triple, and introduced his interesting research project to me in 2008. The spectral triple in his mind turned out to be the pr ..."
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I am very grateful to BaiLing Wang. He foresaw the possibility that the projective spin Dirac operator defined by [19] in formal sense can be realized by a certain spectral triple, and introduced his interesting research project to me in 2008. The spectral triple in his mind turned out to be the projective spectral triple constructed in this paper. Without his insight, I wouldn’t have been writing this thesis. I also wish to thank my advisor, Matilde Marcolli, for her many years of encouragement, support, and many helpful suggestions on both this research and other aspects. iv We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted Khomology of the manifold. This socalled “projective spectral triple ” is Morita equivalent to the wellknown commutative spin spectral triple provided that the manifold is spinc. We give an explicit local formula for the twisted Chern character for Ktheories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincaré dual of the Ahat genus of the manifold.
The Chern character for twisted Ktheory of orbifolds. ArXiv math
, 2005
"... For an orbifold X and α ∈ H 3 (X, Z), we introduce the twisted cohomology H ∗ c (X, α) and prove that the ConnesChern character establishes an isomorphism between the twisted Kgroups K ∗ α(X)⊗C and twisted cohomology H ∗ c (X, α). This theorem, on the one hand, generalizes a classical result of Ba ..."
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For an orbifold X and α ∈ H 3 (X, Z), we introduce the twisted cohomology H ∗ c (X, α) and prove that the ConnesChern character establishes an isomorphism between the twisted Kgroups K ∗ α(X)⊗C and twisted cohomology H ∗ c (X, α). This theorem, on the one hand, generalizes a classical result of BaumConnes, BrylinskiNistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the Kgroups of X tensored with C, and the compactlysupported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of AdemRuan regarding the Chern character isomorphism of twisted orbifold Ktheory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as MathaiStevenson’s theorem regarding the Chern character isomorphism of twisted Ktheory of a compact manifold. Contents
1 THE ENTIRE CYCLIC COHOMOLOGY OF
, 709
"... Abstract. Our aim in this paper is to compute the entire cyclic cohomology of noncommutative 2tori. First of all, we clarify their algebraic structure of noncommutative 2tori as a F ∗algebra, according to the idea of ElliottEvans. Actually, they are the inductive limit of subhomogeneous F ∗alge ..."
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Abstract. Our aim in this paper is to compute the entire cyclic cohomology of noncommutative 2tori. First of all, we clarify their algebraic structure of noncommutative 2tori as a F ∗algebra, according to the idea of ElliottEvans. Actually, they are the inductive limit of subhomogeneous F ∗algebras. Using such a result, we compute their entire cyclic cohomology, which is isomorphic to their periodic one as a complex vector space. 1.
1 THE ENTIRE CYCLIC COHOMOLOGY OF NONCOMMUTATIVE 2TORI
, 709
"... Abstract. We compute the entire cyclic cohomology of the noncommutative 2tori, which are isomorphic to their periodic ones. 1. ..."
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Abstract. We compute the entire cyclic cohomology of the noncommutative 2tori, which are isomorphic to their periodic ones. 1.
Z2graded Čech Cohomology in Noncommutative Geometry ∗ Do Ngoc Diep
, 2008
"... The Z2graded Čech cohomology theory is considered in the framework of noncommutative geometry over complex number field and in particular the homotopy invariance and Morita invariance are proven. In some special case we deduce an isomorphism between this noncommutative theory and the classical Z2g ..."
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The Z2graded Čech cohomology theory is considered in the framework of noncommutative geometry over complex number field and in particular the homotopy invariance and Morita invariance are proven. In some special case we deduce an isomorphism between this noncommutative theory and the classical Z2graded Čech cohomology theory. Keywords: Čech cohomology of a sheaf, cyclic theory