Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2008. / Title from title screen. Includes bibliographical references.

In this dissertation, we explore the approximate diagonalization of unital homomorphisms between C*-algebras. In particular, we prove that unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with tracial rank at most one are approximately diagonalizable. This is equivalent to the approximate diagonalization of commuting sets of normal matrices. We also prove limited generalizations of this theorem. Namely, certain injective unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with rational tracial rank at most one are shown to be approximately diagonalizable. Also unital injective homomorphisms from AH-algebras with unique tracial state into separable simple unital C*-algebras of tracial rank at most one are proved to be approximately diagonalizable. Counterexamples are provided showing that these results cannot be extended in general. Finally, we prove that for unital homomorphisms between AF-algebras, approximate diagonalization is equivalent to a combinatorial problem involving sections of lattice points in cones.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2014. / Made available in DSpace on 2014-08-06T17:56:09Z (GMT). No. of bitstreams: 1 326845.pdf: 667935 bytes, checksum: f31ea3bc71ffcca9a1f9d505f0ddfdbb (MD5) Previous issue date: 2014 / Um grupo quase-reticulado é um tipo especial de grupo parcialmente ordenado. Podemos associar C*-álgebras a grupos quase-reticulados e definir amenabilidade de forma natural. Nosso objetivo neste trabalho é provar um resultado devido a Laca e Raeburn que dá uma caracterização simples para as representações fiéis da C*-álgebra de um grupo quase-reticulado amenable. Apresentaremos também uma aplicação desse teorema.<br> / Abstract : A quasi-lattice group is a special type of partially ordered group. We can associate C*-algebras to quasi-lattice groups and define amenability in a natural way. Our goal in this work is to prove a result due to Laca and Raeburn which gives a simple characterization for the faithful representations of an amenable quasi-lattice group C*-algebra. We also present an application of this theorem.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2013 / Made available in DSpace on 2013-06-26T01:09:14Z (GMT). No. of bitstreams: 1 315200.pdf: 746108 bytes, checksum: ec240ad72ab8b3d84ba36d336deccabb (MD5) / Um fibrado de Fell sobre um grupo discreto G é uma família B de espaços de Banach indexada por G e munida com operações de multiplicação e involução compatíveis com a estrutura do grupo G. Neste trabalho, entre outros resultados, apresentaremos uma equivalência entre a categoria dos fibrados de Fell sobre G e a categoria das coações contínuas e injetivas do grupo quântico compacto reduzido do grupo (obtido da C*-álgebra reduzida do grupo) em C*-álgebras. No caso em que o grupo é abeliano, mostraremos que a categoria dos fibrados sobre G é equivalente à categoria das ações fortemente contínuas do dual de Pontryagin de G em C*-álgebras.<br> / Abstract : A Fell bundle over a discrete group G is a family of Banach spaces {Bt}tEG equipped with multiplication and involution compatible with the group structure of G. In this paper, among others results, we will present an equivalence between the category B of Fell bundles over G and the category of continuous and injective coactions of the compact quantum group C*r (G) in C*-algebras. When the group G is abelian, we will show that B is equivalent to the category of strongly continuous actions of the Pontryagin dual G in C*-algebras.

My research lies in the general area of functional analysis. I am particularly interested in C*-algebras and related dynamical systems. From the very beginning of the theory of operator algebras, in the works of Murray and von Neumann dating from the mid 1930's, dynamical systems and operator algebras have led a symbiotic existence. Murray and von Neumann's work grew from a few esoteric, but clearly original and prescient papers, to a ma jor river of contemporary mathematics. My work lies at the confluence of two important tributaries to this river. On the one hand, the operator algebras that I study are C*-algebras that are built from graphs. On the other, the dynamical systems on which I focus are symbolic dynamical systems of various types. My goal is to use dynamical systems theory to construct new and interesting C*-algebras and to use the algebraic invariants of these algebras to reveal properties of the dynamics. My work has two fairly distinct strands: One deals with C*-algebras built from irreversible dynamical systems. The other deals with group actions on graph C*-algebras and their generalizations.

Orientador: Jorge Tulio Ascui Mujica / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-21T23:57:51Z (GMT). No. of bitstreams: 1 Santos_ElisaReginados_D.pdf: 2025728 bytes, checksum: 9a431cb6242382a2edc9a78d9a7467e4 (MD5) Previous issue date: 2013 / Resumo: O resumo poderá ser visualizado no texto completo da tese digital / Abstract: The abstract is available with the full electronic document / Doutorado / Matematica / Doutor em Matemática

A Cuntz-Pimsner algebra is a quotient of a generalized Toeplitz algebra. It is completely determined by a C*-correspondence, which consists of a right Hilbert A- module, E, and a *-homomorphism from the C*-algebra A into L(E), the adjointable operators on E. Some familiar examples of C*-algebras which can be recognized as Cuntz-Pimsner algebras include the Cuntz algebras, Cuntz-Krieger algebras, and crossed products of a C*-algebra by an action of the integers by automorphisms. In this dissertation, we construct a Cuntz-Pimsner Algebra associated to a dynam- ical system of a substitution tiling, which provides an alternate construction to the groupoid approach found in [3], and has the advantage of yielding a method for com- puting the K-Theory. / Graduate

Trabalhamos com funções definidas em Rn que tomam valores numa C*-álgebra A. Consideramos o conjunto SA (Rn) das funções de Schwartz, (de decrescimento rápido), com norma dada por ||f||2 = ||?f(x)*f(x)dx||½. Denotamos por CB?(R2n,A) o conjunto das funções C? com todas as suas derivadas limitadas. Provamos que os operadores pseudo-diferenciais com símbolo em CB?(R2n,A) são contínuos em SA(Rn) com a norma || ? ||2, fazendo uma generalização de [10]. Rieffel prova em [1] que CB?(Rn,A) age em SA(Rn) por meio de um produto deformado, induzido por uma matriz anti-simétrica, J, como segue: LFg(x)=F×Jg(x) = ?e2?iuvF(x+Ju)g(x+v)dudv, (integral oscilatória). Dizemos que um operador S é Heisenberg-suave se as aplicações z |-> T-zSTz e ? |-> M-?SM?, z,? E Rn, são C? onde Tzg(x)=g(x-z) e M?g(x)=ei?xg(x). No final do capítulo 4 de [1], Rieffel propõe uma conjectura: que todos os operadores \"adjuntáveis\" em SA(Rn), Heisenberg-suaves, que comutam com a representação regular à direita de CB?(Rn,A), RGf = f×JG, são os operadores do tipo LF. Provamos este resultado para o caso A=|C, ver [14], usando a caracterização de Cordes (ver [17]) dos operadores Heisenberg-suaves em L2(Rn) como sendo os operadores pseudo-diferenciais com símbolo em CB?(R2n). Também é provado neste trabalho que, se vale uma generalização natural da caracterização de Cordes, a conjectura de Rieffel é verdadeira. / We work with functions defined on Rn with values in a C*-algebra A. We consider the set SA(Rn) of Schwartz functions (rapidly decreasing), with norm given by ||f||2 = ||?f(x)*f(x)dx||½ . We denote CB?(R2n,A) the set of functions which are C? and have all their derivatives bounded. We prove that pseudo-differential operators with symbol in CB?(R2n,A) are continuous on SA(Rn) with the norm || · ||2, thus generalizing the result in [10]. Rieffel proves in [1] that CB?(Rn,A) acts on SA(Rn) through a deformed product induced by an anti-symmetric matrix, J, as follows: LFg(x)=F×Jg(x) = ?e2?iuvF(x+Ju)g(x+v)dudv (an oscillatory integral). We say that an operator S is Heisenberg-smooth if the maps z |-> T-zSTz and ? |-> M-?SM?, z,? E Rn are C?; where Tzg(x)=g(x-z) and where M?g(x)=ei?xg(x). At the end of chapter 4 of [1], Rieffel proposes a conjecture: that all "adjointable" operators in SA(Rn) that are Heisenberg-smooth and that commute with the right-regular representation of CB?(Rn,A), RGf = f×JG, are operators of type LF . We proved this result for the case A = |C in [14], using Cordes\' characterization of Heisenberg-smooth operators on L2(Rn) as being the pseudo-differential operators with symbol in CB?(R2n). It is also proved in this thesis that, if a natural generalization of Cordes\' characterization is valid, then the Rieffel conjecture is true.

Trabalhamos com funções definidas em Rn que tomam valores numa C*-álgebra A. Consideramos o conjunto SA (Rn) das funções de Schwartz, (de decrescimento rápido), com norma dada por ||f||2 = ||?f(x)*f(x)dx||½. Denotamos por CB?(R2n,A) o conjunto das funções C? com todas as suas derivadas limitadas. Provamos que os operadores pseudo-diferenciais com símbolo em CB?(R2n,A) são contínuos em SA(Rn) com a norma || ? ||2, fazendo uma generalização de [10]. Rieffel prova em [1] que CB?(Rn,A) age em SA(Rn) por meio de um produto deformado, induzido por uma matriz anti-simétrica, J, como segue: LFg(x)=F×Jg(x) = ?e2?iuvF(x+Ju)g(x+v)dudv, (integral oscilatória). Dizemos que um operador S é Heisenberg-suave se as aplicações z |-> T-zSTz e ? |-> M-?SM?, z,? E Rn, são C? onde Tzg(x)=g(x-z) e M?g(x)=ei?xg(x). No final do capítulo 4 de [1], Rieffel propõe uma conjectura: que todos os operadores \"adjuntáveis\" em SA(Rn), Heisenberg-suaves, que comutam com a representação regular à direita de CB?(Rn,A), RGf = f×JG, são os operadores do tipo LF. Provamos este resultado para o caso A=|C, ver [14], usando a caracterização de Cordes (ver [17]) dos operadores Heisenberg-suaves em L2(Rn) como sendo os operadores pseudo-diferenciais com símbolo em CB?(R2n). Também é provado neste trabalho que, se vale uma generalização natural da caracterização de Cordes, a conjectura de Rieffel é verdadeira. / We work with functions defined on Rn with values in a C*-algebra A. We consider the set SA(Rn) of Schwartz functions (rapidly decreasing), with norm given by ||f||2 = ||?f(x)*f(x)dx||½ . We denote CB?(R2n,A) the set of functions which are C? and have all their derivatives bounded. We prove that pseudo-differential operators with symbol in CB?(R2n,A) are continuous on SA(Rn) with the norm || · ||2, thus generalizing the result in [10]. Rieffel proves in [1] that CB?(Rn,A) acts on SA(Rn) through a deformed product induced by an anti-symmetric matrix, J, as follows: LFg(x)=F×Jg(x) = ?e2?iuvF(x+Ju)g(x+v)dudv (an oscillatory integral). We say that an operator S is Heisenberg-smooth if the maps z |-> T-zSTz and ? |-> M-?SM?, z,? E Rn are C?; where Tzg(x)=g(x-z) and where M?g(x)=ei?xg(x). At the end of chapter 4 of [1], Rieffel proposes a conjecture: that all ”adjointable” operators in SA(Rn) that are Heisenberg-smooth and that commute with the right-regular representation of CB?(Rn,A), RGf = f×JG, are operators of type LF . We proved this result for the case A = |C in [14], using Cordes\' characterization of Heisenberg-smooth operators on L2(Rn) as being the pseudo-differential operators with symbol in CB?(R2n). It is also proved in this thesis that, if a natural generalization of Cordes\' characterization is valid, then the Rieffel conjecture is true.

This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [10]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II 1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [59]. We also obtain a product version of this result: any maximal abelian ∗-subalgebra of any II 1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana's cocycle superrigidity theorem [36], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article [11] which has already been submitted for publication.