Browsing by Subject "Banach algebras"
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Item Cluster Value Problems in Infinite-Dimensional Spaces(2014-08-05) Ortega Castillo, SofiaIn this dissertation we study cluster value problems for Banach algebras H(B) of analytic functions on the open unit ball B of a Banach space X that contain X* and 1. Solving cluster value problems requires understanding the cluster set of a function f ? H(B). For the Banach spaces X we focus on, such as those with a shrinking reverse monotone Finite Dimensional Decomposition and C(K), we prove cluster value theorems for a Banach algebra H(B) and a point x** ? B ?**. In doing so, we apply standard methods and results in functional analysis; in particular we use the facts that projections from X onto a finite-codimensional subspace equal I_(X) minus a finite rank operator and that C(K)* = l_(1)(K) when K is compact, Hausdorff and dispersed. We also prove that for any separable Banach space Y , a cluster value problem for H(BY ) (H = H? or H = Au) can be reduced to a cluster value problem for H(BX) for some Banach space X that is an l_(1)-sum of a sequence of finite-dimensional spaces. The proof relies on the construction of an isometric quotient map from a suitable X to Y that induces an isometric algebra homomorphism from H(BY ) to H(BX) with norm one left inverse. The left inverse is built using ultrafilter techniques. Other tools include the infinite-dimensional version of the Schwarz lemma and familiar one complex variable results such as Cauchy's inequality and Montel's theorem. We conclude this work by describing the related ? ? problem and defining strong pseudoconvexity as well as uniform strong pseudoconvexity in the context of Banach spaces. Our last result is that 2-uniformly PL-convex Banach spaces have a uniformly strictly pseudoconvex unit ball. In future research we will study the ? ? problem in uniformly strictly pseudoconvex unit balls and in the open unit ball of finite-dimensional Banach spaces such as the ball of l_1^n.Item Results concerning some convolution algebras(Texas Tech University, 1978-05) Wiggins, Gary LeonThis dissertation presents results concerning the properties of two classes of Banach algebras of integrable functions. The theory of algebras of integrable functions on a topological group is a point of nexus between two of the main branches of mathematlcal analysis, abstract harmonic analysis and the theory of Banach algebras. Abstract harmonlc analysis developed from the classical theory of Fourier series and Fourier Integrals on the circle, the integers, and the real line (28). The process of abstraction began in the 1920's with the explicit definition of the notion of a topological group. The work of Haar (8), von Neumann (22), and Weil (26) established the existence of Haar measure and extended the realm of Fourier analysis to certain classes of topological groups. The main thrust of abstract harmonic analysis has been an effort to interpret and generalize the results of classical Fourier analysis In this abstract setting. Results in this direction are collected in various texts and reference works (11) (12) (17) (24).Item Results relating to multipliers and double centralizers of B*-algebras and certain A*-algebras.(Texas Tech University, 1974-08) Davenport, John WayneNot available