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Twisted KTheory and KTheory of Bundle Gerbes
 COMMUN. MATH. PHYS
, 2002
"... In this note we introduce the notion of bundle gerbe Ktheory and investigate the relation to twisted Ktheory. We provide some examples. Possible applications of bundle gerbe Ktheory to the classification of Dbrane charges in nontrivial backgrounds are briefly discussed. ..."
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Cited by 139 (32 self)
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In this note we introduce the notion of bundle gerbe Ktheory and investigate the relation to twisted Ktheory. We provide some examples. Possible applications of bundle gerbe Ktheory to the classification of Dbrane charges in nontrivial backgrounds are briefly discussed.
Affine buildings, tiling systems and higher rank CuntzKrieger algebras
, 1999
"... Abstract. To an rdimensional subshift of finite type satisfying certain special properties we associate a C∗algebra A. This algebra is a higher rank version of a CuntzKrieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: if Γ is a group acting freely on t ..."
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Cited by 90 (13 self)
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Abstract. To an rdimensional subshift of finite type satisfying certain special properties we associate a C∗algebra A. This algebra is a higher rank version of a CuntzKrieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: if Γ is a group acting freely on the vertices of an Ã2 building, with finitely many orbits, and if Ω is the boundary of that building, then C(Ω)o Γ is the algebra associated to a certain two dimensional subshift.
The Novikov conjecture for groups with finite asymptotic dimension
 Ann. of Math
, 1998
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Twisted Ktheory of differentiable stacks
 ANN. SCI. ÉCOLE NORM. SUP
, 2004
"... In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
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Cited by 74 (13 self)
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In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted Ktheories including the usual twisted Ktheory of topological spaces, twisted equivariant Ktheory, and the twisted Ktheory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted Kgroups can be expressed by socalled “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of Ktheory (KKtheory) of C ∗algebras.
Noncommutative geometry of tilings and gap labelling
 Rev. Math. Phys
, 1995
"... To a given tiling a non commutative space and the corresponding C ∗algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for discrete models of one or many particle systems on the til ..."
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Cited by 54 (13 self)
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To a given tiling a non commutative space and the corresponding C ∗algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for discrete models of one or many particle systems on the tiling or its periodic identification. Its scaled ordered K0group furnishes the gap labelling of Schrödinger operators. The group is computed for one dimensional tilings and Cartesian products thereof. Its image under a state is investigated for tilings which are invariant under a substitution. Part of this image is given by an invariant measure on the hull of the tiling which is determined. The results from the Cartesian products of one dimensional tilings point out that the gap labelling by means of the values of the integrated density of states is already fully determined by this measure.
A universal multicoefficient theorem for the Kasparov groups
 Duke Math. J
, 1996
"... 1. Introduction. Let K(A) denote the sum of all the Ktheory groups of a C*algebra A in all degrees and with all cyclic coefficient groups. The Bockstein operations (which generate a category Λ) act on K(A). We establish a universal coefficient exact sequence 0 → Pext(K∗(A),K∗(B)) δ− → KK(A,B) Γ− → ..."
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Cited by 54 (8 self)
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1. Introduction. Let K(A) denote the sum of all the Ktheory groups of a C*algebra A in all degrees and with all cyclic coefficient groups. The Bockstein operations (which generate a category Λ) act on K(A). We establish a universal coefficient exact sequence 0 → Pext(K∗(A),K∗(B)) δ− → KK(A,B) Γ− → HomΛ(K(A),K(B)) → 0.
The local structure of tilings and their integer group of invariants
 Comm. Math. Phys
, 1997
"... The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the K0group of the groupoid C ∗algebra for ..."
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Cited by 52 (13 self)
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The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the K0group of the groupoid C ∗algebra for tilings which reduce to decorations of Z d. The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and primitive. This yields in particular the set of possible gap labels predicted by Ktheory for Schrödinger operators describing the particle motion in such a tiling. KCLTH956 1
Nonstable Ktheory for graph algebras
 Algebr. Represent. Th. DOI
"... Abstract. We compute the monoid V (LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattic ..."
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Abstract. We compute the monoid V (LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of LK(E) and the lattice of orderideals of V (LK(E)). When K is the field C of complex numbers, the algebra LC(E) is a dense subalgebra of the graph C ∗algebra C ∗ (E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*algebra of any rowfinite graph turns out to satisfy the stable weak cancellation property.
Spaces of tilings, finite telescopic approximations and gaplabelling
"... For a large class of tilings of R d, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull ΩT of such a tiling T inherits a minimal R dlamination structure with flat leaves and a transversal ΓT which is a Cantor set. In this case, we show ..."
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Cited by 49 (3 self)
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For a large class of tilings of R d, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull ΩT of such a tiling T inherits a minimal R dlamination structure with flat leaves and a transversal ΓT which is a Cantor set. In this case, we show that the continuous hull can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact dmanifolds. Truncated sequences furnish better and better finite approximations of the asymptotic dynamical system and the algebraic topological features related to this sequence reflect the dynamical properties of the R daction on the continuous hull. In particular the set of positive invariant measures of this action turns to be a convex cone, canonically associated with the orientation, in the projective limit of the d th homology groups of the branched manifolds. As an application of this construction we prove a gaplabelling theorem: Consider the C ∗algebra AT of ΩT, and the group K0(AT), then for every finite R dinvariant measure µ on ΩT, the image of the group K0(AT) by the µtrace satisfies: Tµ(K0(AT)) = ΓT dµ t C(ΓT,Z), where µ t is the transverse invariant measure on ΓT induced by µ and C(ΓT,Z) the set of continuous functions on ΓT with integer values. 1 1