In type II noncommutative geometry, the geometry on a C∗-algebra A is given by an unbounded Breuer–Fredholm module (ρ,N,D) over A. Here ρ:A→N is a ∗-homomorphism from A to the semi-finite von Neumann algebra N⊂B(H), and D is an unbounded Breuer–Fredholm operator affiliated with N that satisfies certain axioms. Each Breuer–Fredholm module assigns an index to a given element in the K-theory of A. The Breuer–Fredholm index provides a real-valued pairing between the K-homology and the K-theory of A. When N=B(H), a construction of Jaffe-Lesniewski-Osterwalder associates to the module (ρ,N,D) a cocycle in the entire cyclic cohomology group of A for D is theta-summable. The JLO character and the K-theory character intertwine the K-theoretical pairing with the pairing of entire cyclic theory. If (ρ,N,F) is a finitely summable bounded Breuer–Fredholm module, Benameur-Fack defined a cocycle generalizing the Connes's cocycle for bounded Fredholm modules. On the other hand, given a finitely-summable unbounded Breuer–Fredholm module, there is a canonically associated bounded Breuer–Fredholm module. The first main result of this thesis extends the JLO theory to Breuer–Fredholm modules (possibly N does not equal B(H)) in the graded case, and proves that the JLO cocycle and Connes cocycle define the same class in the entire cyclic cohomology of A. This extends a result of Connes-Moscovici for Fredholm modules. An example of an unbounded Breuer–Fredholm module is given by the noncommutative space of G-connections due to Aastrup-Grimstrup-Nest. In their original work, the authors limit their construction to the case that the group G=U(1) or G=SU(2). Another main result of the thesis extends AGN’s construction to any connected compact Lie group G; and generalizes by considering connections defined on sequences of graphs, using limits of spectral triples. Our construction makes it possible to equip the module (ρ,N,D) with a Z_2-grading. The last part of this thesis studies the JLO character of the Breuer–Fredholm module of AGN. The definition of this Breuer–Fredholm module depends on a divergent sequence. A concrete condition on possible perturbations of the sequence ensuring that the resulting JLO class remains invariant is established. The condition implies a certain functoriality of AGN’s construction.

In type II noncommutative geometry, the geometry on a C∗-algebra A is given by an unbounded Breuer–Fredholm module (ρ,N,D) over A. Here ρ:A→N is a ∗-homomorphism from A to the semi-finite von Neumann algebra N⊂B(H), and D is an unbounded Breuer–Fredholm operator affiliated with N that satisfies certain axioms. Each Breuer–Fredholm module assigns an index to a given element in the K-theory of A. The Breuer–Fredholm index provides a real-valued pairing between the K-homology and the K-theory of A. When N=B(H), a construction of Jaffe-Lesniewski-Osterwalder associates to the module (ρ,N,D) a cocycle in the entire cyclic cohomology group of A for D is theta-summable. The JLO character and the K-theory character intertwine the K-theoretical pairing with the pairing of entire cyclic theory. If (ρ,N,F) is a finitely summable bounded Breuer–Fredholm module, Benameur-Fack defined a cocycle generalizing the Connes's cocycle for bounded Fredholm modules. On the other hand, given a finitely-summable unbounded Breuer–Fredholm module, there is a canonically associated bounded Breuer–Fredholm module. The first main result of this thesis extends the JLO theory to Breuer–Fredholm modules (possibly N does not equal B(H)) in the graded case, and proves that the JLO cocycle and Connes cocycle define the same class in the entire cyclic cohomology of A. This extends a result of Connes-Moscovici for Fredholm modules. An example of an unbounded Breuer–Fredholm module is given by the noncommutative space of G-connections due to Aastrup-Grimstrup-Nest. In their original work, the authors limit their construction to the case that the group G=U(1) or G=SU(2). Another main result of the thesis extends AGN’s construction to any connected compact Lie group G; and generalizes by considering connections defined on sequences of graphs, using limits of spectral triples. Our construction makes it possible to equip the module (ρ,N,D) with a Z_2-grading. The last part of this thesis studies the JLO character of the Breuer–Fredholm module of AGN. The definition of this Breuer–Fredholm module depends on a divergent sequence. A concrete condition on possible perturbations of the sequence ensuring that the resulting JLO class remains invariant is established. The condition implies a certain functoriality of AGN’s construction.

A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.

Thesis (Ph. D.)--University of Chicago, Dept. of Physics, Dec. 2003. / Includes bibliographical references. Also available on the Internet.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ian

Thesis (Ph.D.) - University of Queensland, 2005. / Includes bibliography.

Nesta tese estudamos a mecânica quântica não-comutativa na situação não-relativística. Nesse contexto, a expansão-1/N é introduzida e aplicada para alguns potenciais de interesse, como o do oscilador anarmônico e do potencial Coulombiano. A convergência da série é então discutida. Propomos uma versão modificada do potencial Coulombiano nãocomutativo, o qual fornece uma expansão 1/N bem comportada. A seguir, introduzimos um novo conjunto de relações de comutação no espaço-tempo não-comutativo satisfazendo uma álgebra de Heisenberg deformada. A equação de Pauli modificada é usada para o cálculo de correções para a energia, com o uso de teoria da perturbação, no contexto da não-comutatividade dependente do spin. / In this thesis we study non-commutative quantum mechanics in nonrelativistic situation. In this context, the 1/N-expansion is introduced and applied to some potentials of interest as the anharmonic oscillator and the Coulomb potential. The convergence of the serie is discussed. We proposed a modied version of the noncommutative Coulombian potential which provides a well-behaved 1/N expansion. Subsequently, we introduce a new set of noncommutative space-time commutation relations which satisfy a spin dependent nonstandard Heisenberg algebra. Modied Pauli equation is used to calculate corrections to the energy by the use of perturbation theory in the noncommutativity spin-dependent context.

Cette thèse vise à développer des outils d'analyse harmonique non-commutative. Elle porte plus précisément sur les inégalités de Khintchine non-commutatives et les intégrales singulières à valeurs opérateur. La première partie est dédiée à des questions d'interpolation des espaces Lp classiques. On généralise et on énonce de nouvelles caractérisations des espaces interpolés entre espaces Lp. Dans une seconde partie, on démontre une forme des inégalités de Khintchine non-commutatives valides dans tous les espaces interpolés entre espace Lp. Celle-ci permet d’unifier les cas p < 2 et p > 2 ainsi que de traiter les espaces Lp faibles, même pour p = 1 ou 2. En s'appuyant sur la première partie, on caractérise les espaces dans lesquels les formules usuelles pour les inégalités de Khintchine sont valides. Dans une dernière partie, on donne une preuve simplifiée de l'inégalité de type (1,1) faible pour les intégrales singulières non-commutatives, un résultat précédemment obtenu par Parcet. Cette simplification nous permet de retrouver rapidement deux autres résultats connus : la pseudolocalisation Lp et l’inégalité de type faible pour les intégrales singulières non-commutatives dont le noyau est à valeurs dans un espace de Hilbert. / The purpose of this thesis is to develop tools of noncommutative harmonic analysis. More precisely, it deals with noncommutative Khintchine inequalities and operator-valued singular integrals. The first part is dedicated to questions of interpolation between classical Lp-spaces. We generalize and state new characterisations of interpolation spaces between Lp-spaces. In a second part, we introduce a form of the noncommutative Khintchine inequalities which holds in every interpolation space between two Lp-spaces. It enables us to unify the cases p < 2 and p > 2 and to deal with weak Lp-spaces even when p = 1 or 2. By relying on the first part, we characterize spaces in which the usual formulas for Khintchine inequalities hold. In a last part, we give a simplified proof of the weak boundedness of noncommutative singular integrals, a result previously obtained by Parcet. This simplification allows us to recover quickly two results: the Lp pseudolocalisation and the weak type inequality for noncommutative singular integrals associated to Hilbert-valued kernels.

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