The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.

The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.

If b is an inner function and T is the unit circle, then composition with b induces an endomorphism, β, of L1(T) that leaves H1(T) invariant. In this document we investigate the structure of the endomorphisms of B(L2(T)) and B(H2(T)) that implement by studying the representations of L1(T) and H1(T) in terms of multiplication operators on B(L2(T)) and B(H2(T)). Our analysis, which was inspired by the work of R. Rochberg and J. McDonald, will range from the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert C*-modules.

In the present thesis, we investigate representations of Cuntz algebras coming from dilations of row co-isometries. First, we give some general results about such representations. Next, we show that by labeling a random walk, a row co-isometry appears naturally. We give an explicit form for representations that come from such random walks. Then, we give some conditions relating to the reducibility of these representations, exploring how properties of a random walk relate to the Cuntz algebra representation that comes from it

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.

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.

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

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23564号 / 理博第4758号 / 新制||理||1682(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 泉 正己, 教授 COLLINS Benoit Vincent Pierre, 教授 加藤 毅 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Cette thèse de doctorat est consacrée à la cohomologie cyclique périodique des produits croisés généralisés. Ces derniers sont des C*-algèbres construites à partir d'un bimodule hilbertien. Notre étude s'organise en deux axes complémentaires : un résultat général valable pour les produits croisés généralisés lisses à croissance modérée et un résultat spécifique aux variétés de Heisenberg quantiques. Dans un premier temps, nous introduisons une classe de " versions lisses " des produits croisés généralisés, que nous appelons " produits croisés généralisés lisses à croissance modérée ". Notre premier résultat est que sur ces algèbres, les foncteurs k-stables, invariants sous difféotopie et semi-exacts (comme la cohomologie cyclique périodique) donnent naissance à un hexagone exact analogue à la suite de Pimsner-Voiculescu. Pour prouver cette propriété, nous nous appuierons sur les travaux de Cuntz et tout particulièrement sur la notion de contexte de Morita. Dans un second temps, nous illustrons cette construction en l'appliquant aux variétés de Heisenberg quantiques (QHM). En tirant profit de l'action du groupe de Heisenberg H3 sur les QHM, nous construisons des représentants explicites de la K-théorie et de la cohomologie cyclique. Nous pouvons alors effectuer des calculs explicites d'appariements de Chern-Connes. En combinant ces calculs avec la suite exacte à 6 termes de la première partie, nous construisons des bases explicites de la cohomologie cyclique périodique des QHM. Notre second résultat est donc une description relativement complète et totalement explicite de la K-théorie et de la cohomologie cyclique périodique des QHM.

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