In their paper [2] Cuntz, Deninger, and Laca introduced a C*-algebra \mathfrak{T}[R] associated to a number ring R and showed that it was functorial for injective ring homomorphisms and had an interesting KMS-state structure, which they computed directly. Although isomorphic to the Toeplitz algebra of the ax+b-semigroup R⋊R^× of R, their C*-algebra \mathfrak{T}[R] was defined in terms of relations on a generating set of isometries and projections. They showed that a homomorphism φ:\mathfrak{T}[R]→ A is injective if and only if φ is injective on a certain commutative *-subalgebra of \mathfrak{T}[R]. In this thesis we give a direct proof of this result, and go on to show that there is a countable collection of projections which detects injectivity, which allows us to simplify their characterization of faithful representations of \mathfrak{T}[R]. / Graduate / 0405 / jaspar.wiart@gmail.com

Dados A uma C*-álgebra com unidade e \\alpha um *-endomorfismo de A, um operador transferência para o par (A, \\alpha) é uma aplicação linear contínua positiva L: A --> A tal que L(\\alpha(a)b) = a L(b), para todo a, b \\in A. Nestas condições, denotamos por T(A, \\alpha, L) a C*-álgebra universal com unidade gerada por A e um elemento S sujeito às relações Sa = \\alpha(a)S e S*aS = L(a). Uma redundância é definida como o par (a, k) \\in A x \\overline{ASS* A} tal que abS = akS, para todo b \\in A. Neste trabalho definimos a C*-álgebra chamada de produto cruzado como o quociente de T(A, \\alpha, L) pelo ideal bilateral fechado I gerado pelo conjunto das diferenças a-k, para todas as redundâncias (a, k) tais que a \\in \\overline, onde R denota a Im \\alpha. Mostramos que quando \\alpha é injetor com imagem hereditária, então o produto cruzado é isomorfo à C*-álgebra universal com unidade, denotada por U(A, \\alpha), gerada por A e uma isometria T sujeita à relação \\alpha(a) = TaT*, para todo a \\in A. Também mostramos que a álgebra de Cuntz-Krieger O_A pode ser caracterizada como o produto cruzado definido neste trabalho. / Given A a C*-algebra with unit and \\alpha an *-endomorphism of A, a transfer operator for the pair (A, \\alpha) is a continuous positive linear map L: A --> A such that L(\\alpha(a)b) = a L(b), for all a, b \\in A. Under these conditions , we denote by T(A, \\alpha, L) the universal C*-algebra with unit generated by A and an element S subject to the relations Sa = \\alpha(a)S and S*aS = L(a). A redundancy is defined as a pair (a, k) \\in A x \\overline{ASS* A} such that abS = akS, for all b \\in A. In tjis work we define the C*-algebra called crossed-product as the quotient of T(A, \\alpha, L) by the closed two-sided ideal I generated by the set of all differences a-k, for all redundancies (a, k) such that a \\in \\overline, where by R we mean Im \\alpha. We prove that when \\alpha is injective with an hereditary range, then the crossed-product is isomorphic to the universal C*-algebra with unit, which we denote by U(A, \\alpha), generated by A and an isometry T subject to the relation \\alpha(a) = TaT*, for all a \\in A. We also prove that the Cuntz-Krieger algebra O_A can be characterized as the crossed-product we define in this work.

The C*-algebra representation of a physical system provides an ideal backdrop for the study of bipartite entanglement, as a natural definition of separability emerges as a direct consequence of the non-abelian nature of quantum systems under this formulation. The focus of this dissertation is the quantification of entanglement for infinite dimensional systems. The use of Choquet’s theory of boundary integrals allows for an integral representation of the states on a C*-algebra and subsequent adaptation of the Convex Roof Measures to infinite dimensional systems. Another measure of entanglement, known as the Quantum Correlation Coefficient, is also shown to be a valid measure of entanglement in infinite dimensions, by making use of the intimate connection between separability and positive maps. / Dissertation (MSc)--University of Pretoria, 2020. / Physics / MSc / Unrestricted

No description available.

By a representation of a C*-algebra A on a Hilbert space H we mean a morphism : A → L(H). After summing up neccessary knowledge from the theory of Banach and Hilbert spaces and C*-al- gebras we show that for every C*-algebra a representation exists. We describe its structure detiledly and we focus on examining cyclic representations. We find out that cyclic representations relate to the state space. Because every state can be expressed as an integral with respect to an appropriate measure on the states, in is possible to assign a measure on the state space to each cyclic represen- tation. Therefore, we investigate connexion of a representation with this measure as same as with the corresponding state. This leads us to the definition of an orthogonal measure. We find out that its properties relate with certain subalgebras of L(H). At the end we show that for a separable C*-algebra it is possible to express a representation fulfilling suitable assumptions in the form of a direct integral. 1

L'effet Hawking prédit, dans un espace-temps décrivant l'effondrement d'une étoile à symétrie sphérique vers un trou noir de Schwarzschild, qu'un observateur statique, situé à l'infini, observera un flux thermal de particules quantiques à la température de Hawking. La première démonstration mathématique de l'effet Hawking pour des champs quantiques libres est due à Bachelot, dont le travail sur les champs de Klein-Gordon a été ensuite étendu aux champs de Dirac, d'abord par Bachelot lui-même, puis par Melnyk. Ces travaux, placés dans le cadre d'une symétrie sphérique, ont été complétés par Häfner, qui donna une démonstration rigoureuse de l'effet Hawking pour des champs de Dirac, autour d'une étoile s'effondrant vers un trou noir de Kerr. Le but de cette thèse est d'étudier l'effet Hawking non plus dans un modèle de champs quantiques libres, où les problèmes posés se ramènent à l'étude d'équations aux dérivées partielles linéaires, mais dans un modèle de champs de Dirac en interaction. L'interaction est supposée à support compact, statique, et localisée à l'extérieur de l'étoile. Nous choisissons de traiter le cas d'un modèle jouet, dans un espace-temps de dimension 1+1, situation à laquelle on peut se ramener, au moins dans le cas libre, en utilisant la symétrie sphérique du problème. Nous étudions le comportement de champs de fermions de Dirac dans différentes situations : d'abord, pour une observable suivant l'effondrement de l'étoile ; puis pour une observable stationnaire ; enfin, pour une interaction dépendante du temps, localisée près de la surface de l'étoile. Dans chacun de ces cas, nous montrons l'existence de l'effet Hawking et donnons l'état limite correspondant. / The Hawking effect predicts that, in a space- time describing the collapse of a spherically symmetric star to a Schwarzschild black hole, a static observer at infinity sees the Unruh state as a thermal state at Hawking temperature. The first mathematical proof of the Hawking effect, in the original setting of Hawking, is due to Bachelot. His work on Klein-Gordon fields has been extended to Dirac fields, in the first place by Bachelot himself, and by Melnyk after that. Those works, placed in the setup of a spherically symmetric star, have been completed by Häfner, who gave a rigorous proof of the Hawking effect for Dirac fields, outside a star collapsing to a Kerr black hole. The aim of this thesis is to study the Hawking effect not for a model of free quantum fields, in which case the problems can be reduced to studies on linear partial differential equations, but for a model of interacting Dirac fields. The interaction will be considered as a static, compactly-supported interaction, living outside the star. We choose to study a toy model in a 1+1 dimensional space-time. Using the fact that the problem is spherically symetric, one can, at least in the free case, reduce the real problem to this toy model. We study the behavior of Dirac fermions fields in various situations : first, for an observable following the star's collapse ; then, for a static observable ; finally, for a time-dependent interaction, fixed close to the star's boundary. In each of those cases, we show the existence of the Hawking Effect and give the corresponding limit state.

I consider the Hofstadter and the Harper operators, regarded as e ective models for a Bloch electron in a uniform magnetic eld, in the limit of weak and strong eld respectively. For each value of the Fermi energy in a spectral gap, I prove that the corresponding Fermi projectors exhibit a geometric duality, expressed in terms of some vector bundles canonically associated to the projectors. As a corollary, I get a rigorous geometric derivation of the TKNN equations. More generally, I prove that analogous equations hold true for any orthogonal projector in the rational rotation C -algebra, alias the algebra of the (rational) noncommutative torus.

In this thesis we discuss linear dependence of translations which is intimately related to the zero divisor conjecture. We also discuss the square integrable representations of the generalized Wyle-Heisenberg group in 𝑛² dimensions and its relations with Gabor's question from Gabor Analysis in the light of the time-frequency equation. We study the zero divisor conjecture in relation to the reduced 𝐶*-algebras and operator norm 𝐶*-algebras. For certain classes of groups we address the zero divisor conjecture by providing an isomorphism between the the reduced 𝐶*-algebra and the operator norm 𝐶*-algebra. We also provide an isomorphism between the algebra of weak closure and the von Neumann algebra under mild conditions. Finally, we prove some theorems about the injectivity of some spaces as ℂ𝐺 modules for some groups 𝐺. / Master of Science

Dans la présente thèse de doctorat, les C*-algèbres des groupes de Lie connexes réels nilpotents de pas deux et du groupe de Lie SL(2,R) sont caractérisées. En outre, comme préparation à une analyse de sa C*-algèbre, la topologie du spectre du produit semi-direct U(n) x H_n est décrite, où H_n dénote le groupe de Lie de Heisenberg et U(n) le groupe unitaire qui agit sur H_n par automorphismes. Pour la détermination des C*-algèbres de groupes, la transformation de Fourier à valeurs opérationnelles est utilisée pour appliquer chaque C*-algèbre dans l'algèbre de tous les champs d'opérateurs bornés sur son spectre. On doit trouver les conditions que satisfait l'image de cette C*-algèbre sous la transformation de Fourier et l'objectif est de la caractériser par ces conditions. Dans cette thèse, il est démontré que les C*-algèbres des groupes de Lie connexes réels nilpotents de pas deux et la C*-algèbre de SL(2,R) satisfont les mêmes conditions, des conditions appelées «limites duales sous contrôle normique». De cette manière, ces C*-algèbres sont décrites dans ce travail et les conditions «limites duales sous contrôle normique» sont explicitement calculées dans les deux cas. Les méthodes utilisées pour les groupes de Lie nilpotents de pas deux et pour le groupe SL(2,R) sont très différentes l'une de l'autre. Pour les groupes de Lie nilpotents de pas deux, on regarde leurs orbites coadjointes et on utilise la théorie de Kirillov, alors que pour le groupe SL(2,R), on peut mener les calculs plus directement / In this doctoral thesis, the C*-algebras of the connected real two-step nilpotent Lie groups and the Lie group SL(2,R) are characterized. Furthermore, as a preparation for an analysis of its C*-algebra, the topology of the spectrum of the semidirect product U(n) x H_n is described, where H_n denotes the Heisenberg Lie group and U(n) the unitary group acting by automorphisms on H_n. For the determination of the group C*-algebras, the operator valued Fourier transform is used in order to map the respective C*-algebra into the algebra of all bounded operator fields over its spectrum. One has to find the conditions that are satisfied by the image of this C*-algebra under the Fourier transform and the aim is to characterize it through these conditions. In the present thesis, it is proved that both the C*-algebras of the connected real two-step nilpotent Lie groups and the C*-algebra of SL(2,R) fulfill the same conditions, namely the “norm controlled dual limit” conditions. Thereby, these C*-algebras are described in this work and the “norm controlled dual limit” conditions are explicitly computed in both cases. The methods used for the two-step nilpotent Lie groups and the group SL(2,R) are completely different from each other. For the two-step nilpotent Lie groups, one regards their coadjoint orbits and uses the Kirillov theory, while for the group SL(2,R) one can accomplish the calculations more directly

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.