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.

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.

In this dissertation we explore the question of existence of a property of group actions on C*-algebras known as the tracial Rokhlin property. We prove existence of the property in a very general setting as well as specialise the question to specific situations of interest. For every countable discrete elementary amenable group G, we show that there always exists a G-action ω with the tracial Rokhlin property on any unital simple nuclear tracially approximately divisible C*-algebra A. For the ω we construct, we show that if A is unital simple and Z-stable with rational tracial rank at most one and G belongs to the class of countable discrete groups generated by finite and abelian groups under increasing unions and subgroups, then the crossed product A 􏰃ω G is also unital simple and Z-stable with rational tracial rank at most one. We also specialise the question to UHF algebras. We show that for any countable discrete maximally almost periodic group G and any UHF algebra A, there exists a strongly outer product type action α of G on A. We also show the existence of countable discrete almost abelian group actions with the "pointwise" Rokhlin property on the universal UHF algebra. Consequently we get many examples of unital separable simple nuclear C*-algebras with tracial rank zero and a unique tracial state appearing as crossed products.

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.

For a given field F we seek all division algebras over F up to isomorphism. This question was first investigated for division algebras of finite dimension over F by Richard Brauer. We discuss the construction of the Brauer group and some examples. Crossed products and PI algebras are then introduced with a focus on Amitsur's non-crossed product algebra. Finally, we look at some modern results of Bell on the Gelfand-Kirillov dimension of finitely generated algebras over F and the classification of their division subalgebras.

For a given field F we seek all division algebras over F up to isomorphism. This question was first investigated for division algebras of finite dimension over F by Richard Brauer. We discuss the construction of the Brauer group and some examples. Crossed products and PI algebras are then introduced with a focus on Amitsur's non-crossed product algebra. Finally, we look at some modern results of Bell on the Gelfand-Kirillov dimension of finitely generated algebras over F and the classification of their division subalgebras.

No description available.

viii, 107 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation consists of two related parts. In the first portion we use the tracial Rokhlin property for actions of a finite group G on stably finite simple unital C *-algebras containing enough projections. The main results of this part of the dissertation are as follows. Let A be a stably finite simple unital C *-algebra and suppose a is an action of a finite group G with the tracial Rokhlin property. Suppose A has real rank zero, stable rank one, and suppose the order on projections over A is determined by traces. Then the crossed product algebra C * ( G, A, à à ±) also has these three properties. In the second portion of the dissertation we introduce an analogue of the tracial Rokhlin property for C *-algebras which may not have any nontrivial projections called the projection free tracial Rokhlin property . Using this we show that under certain conditions if A is an infinite dimensional simple unital C *-algebra with stable rank one and à à ± is an action of a finite group G with the projection free tracial Rokhlin property, then C * ( G, A, à à ±) also has stable rank one. / Adviser: Phillips, N. Christopher

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.

viii, 113 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation consists of four principal parts. In the first, we introduce the tracial quasi-Rokhlin property for an automorphism α of a C *-algebra A (which is not assumed to be simple or to contain any projections). We then prove that under suitable assumptions on the algebra A , the associated crossed product C *-algebra C *([Special characters omitted.] , A , α) is simple, and the restriction map between the tracial states of C *([Special characters omitted.] , A , α) and the α-invariant tracial states on A is bijective. In the second part, we introduce a comparison property for minimal dynamical systems (the dynamic comparison property) and demonstrate sufficient conditions on the dynamical system which ensure that it holds. The third part ties these concepts together by demonstrating that given a minimal dynamical system ( X, h ) and a suitable simple C *-algebra A , a large class of automorphisms β of the algebra C ( X, A ) have the tracial quasi-Rokhlin property, with the dynamic comparison property playing a key role. Finally, we study the structure of the crossed product C *-algebra B = C *([Special characters omitted.] , C ( X , A ), β) by introducing a subalgebra B { y } of B , which is shown to be large in a sense that allows properties B { y } of to pass to B . Several conjectures about the deeper structural properties of B { y } and B are stated and discussed. / Committee in charge: Christopher Phillips, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Huaxin Lin, Member, Mathematics; Marcin Bownik, Member, Mathematics; Van Kolpin, Outside Member, Economics