Ten years ago, J. Villadsen constructed the first examples of simple C*-algebras with stable rank other than one or infinity. Villadsen's examples all had a unique tracial state. It is natural to ask whether examples can be found of simple C*-algebras with higher stable rank and more than one tracial state; by building on Villadsen's construction, we describe such examples that admit arbitrary tracial state spaces.

Ten years ago, J. Villadsen constructed the first examples of simple C*-algebras with stable rank other than one or infinity. Villadsen's examples all had a unique tracial state. It is natural to ask whether examples can be found of simple C*-algebras with higher stable rank and more than one tracial state; by building on Villadsen's construction, we describe such examples that admit arbitrary tracial state spaces.

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 68-69). Also available for download via the World Wide Web; free to University of Oregon users.

C*-algebras from tilings are of particular interest. In 1998 J. Anderson and I. Putnam introduced a C*-algebra obtained from a substitution tiling that is viewed today as a standard invariant for this tilings. In this thesis we introduce another C*-algebra associated to a substitution tiling. We expect this C*-algebra to be in some sense a dual C*-algebra to the one introduced by Anderson and Putnam. but we were not able to make a precise statement. In our effort to characterize this new C*-algebras we prove that they are simple and can be constructed as an inductive limit of recursive subhomogenous algebras. We finish with K-theory computations for a number of examples.

The abstract Banach associative symmetrical *-algebras over C, so called C*- algebras, were introduced first in 1943 by Gelfand and Naimark24. In the present time the theory of C*-algebras has become a vast portion of functional analysis having connections and applications in almost all branches of modern mathematics and theoretical physics. From the 1940’s and the beginning of 1950’s there were numerous attempts made to extend the theory of C*-algebras to a category wider than Banach algebras. For example, in 1952, while working on the theory of locally-multiplicatively-convex algebras as projective limits of projective families of Banach algebras, Arens in the paper8 and Michael in the monograph48 independently for the first time studied projective limits of projective families of functional algebras in the commutative case and projective limits of projective families of operator algebras in the non-commutative case. In 1971 Inoue in the paper33 explicitly studied topological *-algebras which are topologically -isomorphic to projective limits of projective families of C*-algebras and obtained their basic properties. He as well suggested a name of locally C*-algebras for that category. For the present state of the theory of locally C*-algebras see the monograph of Fragoulopoulou. Also there were many attempts to extend the theory of C*-algebras to nonassociative algebras which are close in properties to associative algebras (in particular, to Jordan algebras). In fact, the real Jordan analogues of C*-algebras, so called JB-algebras, were first introduced in 1978 by Alfsen, Shultz and Størmer in1. One of the main results of the aforementioned paper stated that modulo factorization over a unique Jordan ideal each JB-algebra is isometrically isomorphic to a JC-algebra, i.e. an operator norm closed Jordan subalgebra of the Jordan algebra of all bounded self-adjoint operators with symmetric multiplication acting on a complex Hilbert space. Projective limits of Banach algebras have been studied sporadically by many authors since 1952, when they were first introduced by Arens8 and Michael48. Projective limits of complex C*-algebras were first mentioned by Arens. They have since been studied under various names by Wenjen, Sya Do-Shin, Brooks, Inoue, Schmüdgen, Fritzsche, Fragoulopoulou, Phillips, etc. We will follow Inoue33 in the usage of the name "locally C*-algebras" for these objects. At the same time, in parallel with the theory of complex C*-algebras, a theory of their real and Jordan analogues, namely real C*-algebras and JB-algebras, has been actively developed by various authors. In chapter 2 we present definitions and basic theorems on complex and real C*-algebras, JB-algebras and complex locally C*-algebras to be used further. In chapter 3 we define a real locally Hilbert space HR and an algebra of operators L(HR) (not bounded anymore) acting on HR. In chapter 4 we give new definitions and study several properties of locally C*- and locally JB-algebras. Then we show that a real locally C*-algebra (locally JBalgebra) is locally isometric to some closed subalgebra of L(HR). In chapter 5 we study complex and real Abelian locally C*-algebras. In chapter 6 we study universal enveloping algebras for locally JB-algebras. In chapter 7 we define and study dual space characterizations of real locally C* and locally JB-algebras. In chapter 8 we define barreled real locally C* and locally JB-algebras and study their representations as unbounded operators acting on dense subspaces of some Hilbert spaces. It is beneficial to extend the existing theory to the case of real and Jordan analogues of complex locally C*-algebras. The present thesis is devoted to study such analogues, which we call real locally C*- and locally JB-algebras. / Mathematics / D. Phil. (Mathematics)

abstract: Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout. / Dissertation/Thesis / Masters Thesis Mathematics 2016

Orientadores: Ruy Exel Filho, Jorge Tulio Mujica Ascui / Tese (Doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T01:37:00Z (GMT). No. of bitstreams: 1 Royer_Danilo_D.pdf: 374070 bytes, checksum: c3e7702f1d2d2d102d668d4c0f823deb (MD5) Previous issue date: 2004 / Doutorado / Matematica / Doutor em Matemática

This document contains a proof that Bieberbach groups with finite abelianization are not connective (an E-theoretic property) and then uses this result to provide a characterization of connectivity in the case of Bieberbach groups.

Renormalization procedures for families of dynamical systems have been used to prove many interesting results. Examples of results include that the bifurcation rate for the attractors of an analytic one-parameter family of quadratic-like maps is universal for all such families, unique ergodicity for almost every interval exchange mapping, a unique ergodicity criterion for the vertical translation flow of a flat surface in terms of its ``renormalization dynamics", known as Masur's criterion, and the classification of circle diffeomorphisms up to $C^{\infty}$ conjugation. We introduce renormalization procedures for $C^{*}$-algebras and étale groupoids using the concepts of $C_{0}(X)$-algebras and Morita equivalence for the former, and groupoid bundles and groupoid equivalence, in the sense of Muhly, Renault and Williams, for the latter. We focus on proving analogs to Masur's criterion in both cases using $C^{*}$-algebraic methods. Applying our criterion to our examples of renormalization procedures provides a unique trace criterion for unital AF algebras extending the one provided by Treviño in the setting of flat surfaces and the one provided by Veech in the setting of interval exchange mappings. Also, we recover the old fact that rotation of the circle by an irrational angle is uniquely ergodic, and the new fact that interesting groupoids associated to certain iterated function systems, recently introduced by Korfanty, have unique invariant probability measures whenever they are minimal. Lastly, we show how an étale groupoid renormalization procedure arises from an étale groupoid which factors down onto a groupoid associated to its renormalization dynamics, whenever it is a local homeomorphism. / Graduate

In this dissertation we develop aspects of ergodic theory for C*-dynamical systems for which the C*-algebras are allowed to be noncommutative. We define four ergodic properties, with analogues in classic ergodic theory, and study C*-dynamical systems possessing these properties. Our analysis will show that, as in the classical case, only certain combinations of these properties are permissable on C*-dynamical systems. In the second half of this work, we construct concrete noncommutative C*-dynamical systems having various permissable combinations of the ergodic properties. This shows that, as in classical ergodic theory, these ergodic properties continue to be meaningful in the noncommutative case, and can be useful to classify and analyse C*-dynamical systems. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mathematics and Applied Mathematics / unrestricted