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393
The quantum structure of spacetime at the Planck scale and quantum
, 1995
"... Abstract. We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg’s principle and by Einstein’s theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. ..."
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Cited by 226 (4 self)
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Abstract. We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg’s principle and by Einstein’s theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS 2 of the 2–sphere. The relations with Connes’ theory of the standard model will be studied elsewhere. 1.
Spaces over a Category and Assembly Maps in Isomorphism Conjectures in Kand LTheory
"... : We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete ..."
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Cited by 49 (12 self)
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: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. Key words: Algebraic K and Ltheory, BaumConnes Conjecture, assembly maps, spaces and spectra over a category AMSclassification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left Gsets, and morphisms are Gmaps. This is a useful construct for o...
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
"... ..."
Rieffel induction as generalized quantum MarsdenWeinstein reduction
 J. Geom. Phys
, 1995
"... reduction ..."
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 42 (12 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.
A categorical approach to imprimitivity theorems for C ∗ dynamical systems
 Mem. Amer. Math. Soc
"... Abstract. Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossedproduct C ∗algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green’s Imprimitivity Theorem for actions of groups, and Mansfield’s Imprimitiv ..."
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Cited by 37 (21 self)
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Abstract. Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossedproduct C ∗algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green’s Imprimitivity Theorem for actions of groups, and Mansfield’s Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossedproduct functors among certain equivariant categories. The categories involved have C ∗algebras with actions or coactions (or both) of a fixed locally compact group G as their objects, and equivariant equivalence classes of rightHilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules. The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction
Jensen’s operator inequality
 Bulletin of London Mathematical Society
"... Abstract. We establish what we consider to be the definitive versions of Jensen’s operator inequality and Jensen’s trace inequality for real functions defined on an interval. This is accomplished by the introduction of genuine noncommutative convex combinations of operators, as opposed to the contr ..."
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Cited by 34 (6 self)
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Abstract. We establish what we consider to be the definitive versions of Jensen’s operator inequality and Jensen’s trace inequality for real functions defined on an interval. This is accomplished by the introduction of genuine noncommutative convex combinations of operators, as opposed to the contractions considered in earlier versions of the theory, [9] & [3]. As a consequence, we no longer need to impose conditions on the interval of definition. We show how this relates to the pinching inequality of Davis [4], and how Jensen’s trace inequality generalizes to C ∗ −algebras. 1.