Noncommutative Borsuk-Ulam Theorems

The Borsuk-Ulam theorem in algebraic topology shows that there are significant restrictions on how any topological sphere interacts with the antipodal action of reflection through the origin (which maps x to -x). For example, any map f from a sphere to itself which is continuous and odd (f(-x) = -f(x)) must be homotopically nontrivial. We consider various equivalent forms of the theorem in terms of the function algebras on spheres and examine which forms generalize to certain noncommutative Banach and C*-algebras with finite group actions. Chapter 1 contains background material on C*-algebras, K-theory, and group actions. Next, in Chapter 2, we examine statements related to the Borsuk-Ulam theorem that may be applied on Banach algebras with actions of the two element group; this work indicates when roots of elements do not exist and is motivated by the results of Ali Taghavi. We see that a variant of the Borsuk-Ulam theorem on the function algebra of a sphere, written in terms of individual odd elements in the algebra, does not extend to the noncommutative setting. In Chapter 3, we show that antipodally equivariant maps between theta-deformed spheres of the same dimension are nontrivial on K-theory. This generalizes the commutative case and parallels the work of Makoto Yamashita on the q-spheres, although our methods are quite different. Finally, Chapter 4 concerns a conjecture of Ludwik Dabrowski that seeks to generalize noncommutative Borsuk-Ulam theory to arbitrary C*-algebras through the use of unreduced suspensions. We prove Dabrowski\u27s conjecture and propose a new direction for continued study

Introduction to Normed *-Algebras and their Representations, 7th ed

This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C*-algebra of a general normed *-algebra

Quantum-Classical Hybrid Systems and their Quasifree Transformations

We study continuous variable systems, in which quantum and classical degrees of freedom are combined and treated on the same footing. Thus all systems, including the inputs or outputs to a channel, may be quantum-classical hybrids. This allows a unified treatment of a large variety of quantum operations involving measurements or dependence on classical parameters. The basic variables are given by canonical operators with scalar commutators. Some variables may commute with all others and hence generate a classical subsystem. We systematically study the class of "quasifree" operations, which are characterized equivalently either by an intertwining condition for phase-space translations or by the requirement that, in the Heisenberg picture, Weyl operators are mapped to multiples of Weyl operators. This includes the well-known Gaussian operations, evolutions with quadratic Hamiltonians, and "linear Bosonic channels", but allows for much more general kinds of noise. For example, all states are quasifree. We sketch the analysis of quasifree preparation, measurement, repeated observation, cloning, teleportation, dense coding, the setup for the classical limit, and some aspects of irreversible dynamics, together with the precise salient tradeoffs of uncertainty, error, and disturbance. Although the spaces of observables and states are infinite dimensional for every non-trivial system that we consider, we treat the technicalities related to this in a uniform and conclusive way, providing a calculus that is both easy to use and fully rigorous.Comment: 63 pages, 6 figure

Quantum-Classical hybrid systems and their quasifree transformations

The focus of this work is the description of a framework for quantum-classical hybrid systems. The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case. Here, we are going to solve two main tasks: The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure. Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined. We start with a general introduction to operator algebras and algebraic quantum theory. Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems. Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism. The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system. Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz: We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system. Then, we present solutions for our initial tasks. We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem. After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra. While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose. They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture. The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity. We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations. All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected

Quantum-Classical Hybrid Systems and their Quasifree Transformations

We study continuous variable systems, in which quantum and classical degrees of freedom are combined and treated on the same footing. Thus all systems, including the inputs or outputs to a channel, may be quantum-classical hybrids. This allows a unified treatment of a large variety of quantum operations involving measurements or dependence on classical parameters. The basic variables are given by canonical operators with scalar commutators. Some variables may commute with all others and hence generate a classical subsystem. We systematically study the class of "quasifree" operations, which are characterized equivalently either by an intertwining condition for phase-space translations or by the requirement that, in the Heisenberg picture, Weyl operators are mapped to multiples of Weyl operators. This includes the well-known Gaussian operations, evolutions with quadratic Hamiltonians, and "linear Bosonic channels", but allows for much more general kinds of noise. For example, all states are quasifree. We sketch the analysis of quasifree preparation, measurement, repeated observation, cloning, teleportation, dense coding, the setup for the classical limit, and some aspects of irreversible dynamics, together with the precise salient tradeoffs of uncertainty, error, and disturbance. Although the spaces of observables and states are infinite dimensional for every non-trivial system that we consider, we treat the technicalities related to this in a uniform and conclusive way, providing a calculus that is both easy to use and fully rigorous

Boundary Actions and C*-Algebraic Properties of Discrete Groups

Η παρούσα εργασία αφορά την αλληλεπίδραση μεταξύ δυναμικών και C*-αλγεβρικών ιδιοτήτων διακριτών ομάδων. Πιο συγκεκριμένα, αν Γ είναι μια διακριτή ομάδα, εξετάζονται χαρακτηρισμοί της C*-απλότητας, της ιδιότητας μοναδικού ίχνους, και της ακρίβειας, οι οποίοι συνδέουν τις ιδιότητες αυτές με τον τρόπο που η Γ δρα στο καθολικό τοπολογικό της σύνορο, όπως αυτό ορίστηκε από τον Furstenberg. Ακρογωνιαίος λίθος των χαρακτηρισμών αυτών είναι η ταύτιση μεταξύ του συνόρου του Furstenberg και του συνόρου του Hamana της Γ, τοπολογικού χώρου που προκύπτει στη θεωρία των Γ-εμφυτευτικών καλυμμάτων.This work is concerned with the interplay between dynamical and C*-algebraic properties of discrete groups. More specifically, for a discrete group Γ, characterisations of C*-simplicity, the unique trace property, and exactness are given, relating these properties with the way Γ acts on its universal topological boundary, in the sense of Furstenberg. The cornerstone of these characterisations is the identification between the Furstenberg boundary and the Hamana boundary of Γ, a topological space arising in the theory of Γ-injective envelopes