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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.

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ABSTRACT. – In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S 1-gerbe 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 K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C ∗-algebras. © 2004 Elsevier SAS RÉSUMÉ. – Dans cet article, nous développons la K-théorie tordue pour les champs différentiables, où la torsion s’effectue par une S 1-gerbe sur le champ en question. Nous en établissons les propriétés générales telles que les suites exactes de Mayer–Vietoris, la périodicité de Bott, et le produit K i α ⊗ K j β → Ki+j α+β. Notre approche fournit un cadre général permettant d’étudier diverses K-théories tordues, en particulier la K-théorie tordue usuelle des espaces topologiques, la K-théorie tordue équivariante, et la K-théorie tordue des orbifolds. Nous donnons également une définition équivalente utilisant des opérateurs de Fredholm, et nous discutons les conditions sous lesquelles les groupes de K-théorie tordue peuvent être réalisés à partir de “fibrés vectoriels tordus”. Notre approche consiste à travailler sur les réalisations concrètes des champs, à savoir les groupoïdes, et s’appuie de façon importante sur les techniques de K-théorie (KK-théorie) des C ∗-algèbres.

: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory 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 L-theory, Baum-Connes Conjecture, assembly maps, spaces and spectra over a category AMS-classification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left G-sets, and morphisms are G-maps. This is a useful construct for o...

Abstract. Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product 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 crossed-product 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 right-Hilbert 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

We give a useful new characterization of the set of all completely positive, trace-preserving maps Φ: M2 → M2 from which one can easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points which can be characterized as having exactly two images on the Bloch sphere. We also discuss a number of related issues about the images and the geometry of the set of stochastic maps, and show that any stochastic map on M2 can be written as a convex combination of two “generalized ” extreme points.

Abstract. Given a group G, we construct, in a canonical way, an inverse semigroup S(G) associated to G. The actions of S(G) areshowntobein one-to-one correspondence with the partial actions of G, bothinthecaseof actions on a set, and that of actions as operators on a Hilbert space. In other words, G and S(G) have the same representation theory. We show that S (G) governs the subsemigroup of all closed linear subspaces of a G-graded C ∗-algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A “partial ” version of the group C ∗-algebra of a discrete group is introduced. While the usual group C ∗-algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C ∗-algebra of the two commutative groups of order four, namely Z/4Z and Z/2Z ⊕ Z/2Z, are not isomorphic. 1.

Abstract. Let G be a countable discrete group which acts isometrically and metrically properly on an infinite-dimensional Euclidean space. We calculate the K-theory groups of the C ∗-algebras C ∗ max(G) andC ∗ red(G). Our result is in accordance with the Baum-Connes conjecture. 1.