In Chapter 0, we give a brief discussion of positive linear functionals on C^*-algebras. We discuss the relation between states and representations of C^*-algebras in section 0.1. Some properties of factorial states and primal ideals of C^*-algebras are considered in section 0.2. In Chapter 1, we determine the primal ideals in certain C^*-algebras. We study an antiliminal C^*-algebra considered by Vesterstro m [38] in section 1.1. A variant of a nonliminal postliminal C^*-algebra constructed by Kadison, Lance and Ringrose ([18] and [3]) is considered in section 1.2. Also, we study in section 1.3 a liminal C^*-algebra constructed by J. Dixmier [10]. Chapter 2 is concerned with tensor products and primal ideals of C^*-algebras. In Chapter 3, we try to answer the following question 'when can the pure state space overlineP(A) be written as a union of weak^*-closed simplicial faces of the quasi state space Q(A)?' In section 3.1, we define a condition which we call (*), in terms of equivalent pure states, and we prove that it is equivalent to the pure state space overlineP(A) being a union of weak^*-closed simplicial faces of the state space S(A), where A is a unital C^*-algebra. We prove that the latter condition is equivalent to the pure state space overlineP(A) being a union of weak^*-closed simplicial faces of the quasi state space Q(A). In section 3.2, we consider questions of stability for the condition (*). Some C^*-algebras whose irreducible representations are of finite dimension are studied in section 3.3. Their pure state spaces are determined, thus giving examples and counter examples in connection with condition (*). Finally, in section 3.4, we prove that the factorial state space overlineF(A) is a union of weak^*-closed simplicial faces of Q(A) if, and only if, A is abelian.

In this thesis, We investigate the problem of when a C*-algebra is commutative through continuous functional calculus, The principal results are that: (1) A C*-algebra A is commutative if and only if e^(ix)e^(iy)=e^(iy)e^(ix), for all self-adjoint elements x,y in A. (2) A C*-algebra A is commutative if and only if e^(x)e^(y)=e^(y)e^(x) for all positive elements x,y in A. We will give an extension of (2) as follows: Let f:[a,b]-->[c,d] be any continuous strictly monotonic function where a,b,c,d in R, a<b,c<d. Then a C*-algebra A is commutative if and only if f(x)f(y)=f(y)f(x), for all self-adjoint elements x,y in A with spec(x) in [a,b] and spec(y) in [a,b].

A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. My thesis deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra A that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over A are finite, while A has no bounded quasitrace. The second example is a non-unital, simple C*-algebra B that is stably non-stable, i.e. no matrix algebra over B is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra B has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.

A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. My thesis deals with the question whether the Blackadar-Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. For suitably well-behaved C*-algebras there is a positive result, but none of the non-unital versions holds in full generality. Two examples of C*-algebras are constructed. The first one is a non-unital, stably commutative C*-algebra A that contradicts the weakest possible generalization of the Blackadar-Handelman theorem: The multiplier algebras of all matrix algebras over A are finite, while A has no bounded quasitrace. The second example is a non-unital, simple C*-algebra B that is stably non-stable, i.e. no matrix algebra over B is a stable C*-algebra. In fact, the multiplier algebras over all matrix algebras of this C*-algebra are not properly infinite. Moreover, the C*-algebra B has no bounded quasitrace and therefore gives a simple counterexample to a possible generalization of the Blackadar-Handelman theorem.

Neste trabalho, apresentamos uma introdução às C*-álgebras e a construção do produto cruzado $A times_{\\alpha} Z$, onde A é uma C*-álgebra com unidade, e $\\alpha$ é um automorfismo em A. Apresentamos, também, uma generalização do Teorema de Fejér, no contexto de produto cruzado. A título de exemplo de produto cruzado, provamos que $C times_ Z$ é isomorfo a C(S^1). Sendo X uma compactificação de Z pela adição dos símbolos $+\\infty$ e $-\\infty$, provamos que o produto cruzado $C(X) times_{\\alpha} Z$ é isomorfo A, o fecho do conjunto dos operadores pseudodiferenciais clássicos de ordem 0 sobre S^1, onde é definido pelo deslocamento. Com posse destes isomorfismos, vimos a implicação da generalização do Teorema de Fejér para C(S^1) e para A. / We present an introduction to C * -algebras and the construction of the crossed product $A times_{\\alpha} Z$, where A is a C *-algebra with unit, and $\\alpha$ is an automorphism in A. We also study a generalization of Fejérs theorem on crossed product context. As an example of crossed product, we prove that $C times_ Z$ is isomorphic to C(S^1). Let X be a compactification of Z by addition of the symbols $+\\infty$ and $-\\infty$. We prove that $C(X) times_{\\alpha} Z$ is isomorphic A, the closure of set of classics pseudo-differential operators of order 0 on S^1, where is defined by a shift. Based on these isomorphisms, we see the implication of the generalization of Fejérs theorem for C(S^1) and A.

Neste trabalho, apresentamos uma introdução às C*-álgebras e a construção do produto cruzado $A times_{\\alpha} Z$, onde A é uma C*-álgebra com unidade, e $\\alpha$ é um automorfismo em A. Apresentamos, também, uma generalização do Teorema de Fejér, no contexto de produto cruzado. A título de exemplo de produto cruzado, provamos que $C times_ Z$ é isomorfo a C(S^1). Sendo X uma compactificação de Z pela adição dos símbolos $+\\infty$ e $-\\infty$, provamos que o produto cruzado $C(X) times_{\\alpha} Z$ é isomorfo A, o fecho do conjunto dos operadores pseudodiferenciais clássicos de ordem 0 sobre S^1, onde é definido pelo deslocamento. Com posse destes isomorfismos, vimos a implicação da generalização do Teorema de Fejér para C(S^1) e para A. / We present an introduction to C * -algebras and the construction of the crossed product $A times_{\\alpha} Z$, where A is a C *-algebra with unit, and $\\alpha$ is an automorphism in A. We also study a generalization of Fejérs theorem on crossed product context. As an example of crossed product, we prove that $C times_ Z$ is isomorphic to C(S^1). Let X be a compactification of Z by addition of the symbols $+\\infty$ and $-\\infty$. We prove that $C(X) times_{\\alpha} Z$ is isomorphic A, the closure of set of classics pseudo-differential operators of order 0 on S^1, where is defined by a shift. Based on these isomorphisms, we see the implication of the generalization of Fejérs theorem for C(S^1) and A.

abstract: C*-algebras of categories of paths were introduced by Spielberg in 2014 and generalize C*-algebras of higher rank graphs. An approximately finite dimensional (AF) C*-algebra is one which is isomorphic to an inductive limit of finite dimensional C*-algebras. In 2012, D.G. Evans and A. Sims proposed an analogue of a cycle for higher rank graphs and show that the lack of such an object is necessary for the associated C*-algebra to be AF. Here, I give a class of examples of categories of paths whose associated C*-algebras are Morita equivalent to a large number of periodic continued fraction AF algebras, first described by Effros and Shen in 1980. I then provide two examples which show that the analogue of cycles proposed by Evans and Sims is neither a necessary nor a sufficient condition for the C*-algebra of a category of paths to be AF. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2020

Thirty years ago, Effros and Shen classified the simple dimension groups with rank two. Every such group is parametrized by an irrational number, and can be constructed as an inductive limit using that number's continued fraction expansion. There is a natural generalization of continued fractions to higher dimensions, and this invites the following question: What dimension groups correspond to multidimensional continued fractions? We describe this class of groups and show how some properties of a continued fraction are reflected in the structure of its dimension group. We also consider a related issue: an Effros-Shen group has been shown to arise in a natural way from the tail equivalence relation on a certain sequence space. We describe a more general class of sequence spaces to which this construction can be applied to obtain other dimension groups, including dimension groups corresponding to multidimensional continued fractions.

No description available.

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.