1 
Exponential dichotomy of ode'sAlHanna, Nada Farid. January 2009 (has links)
Thesis (M.S.)University of Texas at El Paso, 2009. / Title from title screen. Vita. CDROM. Includes bibliographical references. Also available online.

2 
The Fourier algebra of a locally trivial groupoidMarti Perez, Laura Raquel January 2011 (has links)
The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its FourierStieltjes algebra B(G) and its Fourier algebra A(G) were defined by Eymard in 1964. Since then, a rich theory has been developed. For the groupoid case, the algebras B(G) and A(G) have been studied by Ramsay and Walter (borelian case, 1997), Renault (measurable case, 1997) and Paterson (locally compact case, 2004). In this work, we present a new definition of A(G) in the locally compact case, specially well suited for studying locally trivial groupoids.
Let G be a locally compact proper groupoid. Following the group case, in order to define A(G), we consider the closure under certain norm of the span of the left regular GHilbert bundle coefficients. With the norm mentioned above, the space A(G) is a commutative Banach algebra of continuous functions of G vanishing at infinity. Moreover, A(G) separates points and it is also a B(G)bimodule. If, in addition, G is compact, then B(G) and A(G) coincide. For a locally trivial groupoid G we present an easier to handle definition of A(G) that involves "trivializing" the left regular bundle.
The main result of our work is a decomposition of A(G), valid for transitive, locally trivial groupoids with a "nice" Haar system. The condition we require the Haar system to satisfy is to be compatible with the Haar measure of the isotropy group at a fixed unit u. If the groupoid is transitive, locally trivial and unimodular, such a Haar system always can be constructed.
For such groupoids, our theorem states that A(G) is isomorphic to the Haagerup tensor product of the space of continuous functions on Gu vanishing at infinity, times the Fourier algebra of the isotropy group at u, times space of continuous functions on Gu vanishing at infinity. Here Gu denotes the elements of the groupoid that have range u. This decomposition provides an operator space structure for A(G) and makes this space a completely contractive Banach algebra.
If the locally trivial groupoid G has more than one transitive component, that we denote Gi, since these components are also topological components, there is a correspondence between GHilbert bundles and families of GiHilbert bundles. Thanks to this correspondence, the FourierStieltjes and Fourier algebra of G can be written as sums of the algebras of the Gi components.
The theory of operator spaces is the main tool used in our work. In particular, the many properties of the Haagerup tensor product are of vital importance.
Our decomposition can be applied to (trivially) locally trivial groupoids of the form X times X and X times H times X, for a locally compact space X and a locally compact group H. It can also be applied to transformation group groupoids X times H arising from the action of a Lie group H on a locally compact space X and to the fundamental groupoid of a pathconnected manifold.

3 
The Fourier algebra of a locally trivial groupoidMarti Perez, Laura Raquel January 2011 (has links)
The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its FourierStieltjes algebra B(G) and its Fourier algebra A(G) were defined by Eymard in 1964. Since then, a rich theory has been developed. For the groupoid case, the algebras B(G) and A(G) have been studied by Ramsay and Walter (borelian case, 1997), Renault (measurable case, 1997) and Paterson (locally compact case, 2004). In this work, we present a new definition of A(G) in the locally compact case, specially well suited for studying locally trivial groupoids.
Let G be a locally compact proper groupoid. Following the group case, in order to define A(G), we consider the closure under certain norm of the span of the left regular GHilbert bundle coefficients. With the norm mentioned above, the space A(G) is a commutative Banach algebra of continuous functions of G vanishing at infinity. Moreover, A(G) separates points and it is also a B(G)bimodule. If, in addition, G is compact, then B(G) and A(G) coincide. For a locally trivial groupoid G we present an easier to handle definition of A(G) that involves "trivializing" the left regular bundle.
The main result of our work is a decomposition of A(G), valid for transitive, locally trivial groupoids with a "nice" Haar system. The condition we require the Haar system to satisfy is to be compatible with the Haar measure of the isotropy group at a fixed unit u. If the groupoid is transitive, locally trivial and unimodular, such a Haar system always can be constructed.
For such groupoids, our theorem states that A(G) is isomorphic to the Haagerup tensor product of the space of continuous functions on Gu vanishing at infinity, times the Fourier algebra of the isotropy group at u, times space of continuous functions on Gu vanishing at infinity. Here Gu denotes the elements of the groupoid that have range u. This decomposition provides an operator space structure for A(G) and makes this space a completely contractive Banach algebra.
If the locally trivial groupoid G has more than one transitive component, that we denote Gi, since these components are also topological components, there is a correspondence between GHilbert bundles and families of GiHilbert bundles. Thanks to this correspondence, the FourierStieltjes and Fourier algebra of G can be written as sums of the algebras of the Gi components.
The theory of operator spaces is the main tool used in our work. In particular, the many properties of the Haagerup tensor product are of vital importance.
Our decomposition can be applied to (trivially) locally trivial groupoids of the form X times X and X times H times X, for a locally compact space X and a locally compact group H. It can also be applied to transformation group groupoids X times H arising from the action of a Lie group H on a locally compact space X and to the fundamental groupoid of a pathconnected manifold.

4 
Operator Ideals in Lipschitz and Operator Spaces CategoriesChavez Dominguez, Javier 2012 August 1900 (has links)
We study analogues, in the Lipschitz and Operator Spaces categories, of several classical ideals of operators between Banach spaces. We introduce the concept of a Banachspacevalued molecule, which is used to develop a duality theory for several nonlinear ideals of operators including the ideal of Lipschitz psumming operators and the ideal of factorization through a subset of a Hilbert space. We prove metric characterizations of pconvex operators, and also of those with Rademacher type and cotype. Lipschitz versions of pconvex and pconcave operators are also considered. We introduce the ideal of Lipschitz (q,p)mixing operators, of which we prove several characterizations and give applications. Finally the ideal of completely (q,p)mixing maps between operator spaces is studied, and several characterizations are given. They are used to prove an operator space version of Pietsch's composition theorem for psumming operators.

5 
Complemented and uncomplemented subspaces of Banach spacesVuong, Thi Minh Thu January 2006 (has links)
"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as wellknown classical sequence spaces (finding nontrivial twisted sums)." Abstract. / Master of Mathematical Sciences

6 
Complemented and uncomplemented subspaces of Banach spacesVuong, Thi Minh Thu . University of Ballarat. January 2006 (has links)
"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as wellknown classical sequence spaces (finding nontrivial twisted sums)." Abstract. / Master of Mathematical Sciences

7 
Results on twisted sums of Banach and operator spaces / Resultados de somas torcidas de espaços de Banach e espaços de operadoresCorrêa, Willian Hans Goes 26 February 2018 (has links)
In this work we study twisted sums induced by complex interpolation, of Banach spaces as well as of operator spaces. In the first part of the thesis we focus on Banach spaces, and clarify how interpolation of families, as of couples, induces an extension of the interpolation space, called the derived space. We study how the types and cotypes of the spaces being interpolated determine the triviality or singularity of the derived space, and we apply the results to the study of submodules of the Schatten classes and in the obtainment of nontrivial twisted sums in which all of the three spaces in the short exact sequence do not have the approximation property. In the second part we develop the theory of twisted sums in the category of operator spaces and present many examples of twisted sums which are completely singular and completely nontrivial. In particular, we solve two versions of Palais\' problem for operator spaces. / Neste trabalho estudamos somas torcidas induzidas por interpolação complexa, tanto de espaços de Banach como de espaços de operadores. Na primeira parte da tese focamos em espaços de Banach, e esclarecemos como a interpolação de famílias, assim como a de pares, gera uma extensão do espaço interpolado, chamada de espaço derivado. Estudamos como os tipos e cotipos dos espaços sendo interpolados influenciam na trivialidade ou singularidade do espaço derivado, e aplicamos os resultados para o estudo de submódulos das classes de Schatten e para a obtenção de somas torcidas nãotriviais em que os três espaços da sequência exata curta não possuem a propriedade da aproximação. Na segunda parte, desenvolvemos a teoria de somas torcidas na categoria de espaços de operadores, e apresentamos vários exemplos de somas torcidas completamente singulares e completamente nãotriviais nessa categoria. Em particular, resolvemos duas versões do problema de Palais para espaços de operadores.

8 
Results on twisted sums of Banach and operator spaces / Resultados de somas torcidas de espaços de Banach e espaços de operadoresWillian Hans Goes Corrêa 26 February 2018 (has links)
In this work we study twisted sums induced by complex interpolation, of Banach spaces as well as of operator spaces. In the first part of the thesis we focus on Banach spaces, and clarify how interpolation of families, as of couples, induces an extension of the interpolation space, called the derived space. We study how the types and cotypes of the spaces being interpolated determine the triviality or singularity of the derived space, and we apply the results to the study of submodules of the Schatten classes and in the obtainment of nontrivial twisted sums in which all of the three spaces in the short exact sequence do not have the approximation property. In the second part we develop the theory of twisted sums in the category of operator spaces and present many examples of twisted sums which are completely singular and completely nontrivial. In particular, we solve two versions of Palais\' problem for operator spaces. / Neste trabalho estudamos somas torcidas induzidas por interpolação complexa, tanto de espaços de Banach como de espaços de operadores. Na primeira parte da tese focamos em espaços de Banach, e esclarecemos como a interpolação de famílias, assim como a de pares, gera uma extensão do espaço interpolado, chamada de espaço derivado. Estudamos como os tipos e cotipos dos espaços sendo interpolados influenciam na trivialidade ou singularidade do espaço derivado, e aplicamos os resultados para o estudo de submódulos das classes de Schatten e para a obtenção de somas torcidas nãotriviais em que os três espaços da sequência exata curta não possuem a propriedade da aproximação. Na segunda parte, desenvolvemos a teoria de somas torcidas na categoria de espaços de operadores, e apresentamos vários exemplos de somas torcidas completamente singulares e completamente nãotriviais nessa categoria. Em particular, resolvemos duas versões do problema de Palais para espaços de operadores.

9 
Lipschitz and commutator estimates, a unified approachPotapov, Denis Sergeevich, January 2007 (has links)
Thesis (Ph.D.)Flinders University, School of Informatics and Engineering, Dept. of Mathematics. / Typescript bound. Includes bibliographical references: (leaves 135140) and index. Also available online.

10 
On the Abstract Structure of Operator Systems and Applications to Quantum Information TheoryRoy M Araiza (10723929) 05 May 2021 (has links)
We introduce the notion of an abstract projection in an operator system and when a finite number of positive contractions in an operator system are all simultaneously abstract projections in that operator system. We extend this notion to Archimedean order unit spaces where we prove when a positive contraction is an abstract projection in some operator system, and furthermore when a finite number of positive contractions in an Archimedean order unit space are all simultaneously abstract projections in a single operator system. These methods are then used to provide new characterizations of both nonsignalling and quantum commuting correlations. In particular, we construct a universal Archimedean order unit space such that every quantum commuting correlation may be realized as the image of a unital linear positive map acting on the generators of that Archimedean order unit space. We also construct an Archimedean order unit space which is universal (in the same way) to nonsignalling correlations. We conclude with results concerning weak dual matrix ordered *vector spaces and the operator systems they induce.

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