Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciencias Fisicas e Matematicas, Curso de Pós-Graduação em Matemática, Florianópolis, 1980 / Made available in DSpace on 2012-10-16T20:21:01Z (GMT). No. of bitstreams: 0Bitstream added on 2016-01-08T13:50:58Z : No. of bitstreams: 1 101381.pdf: 36230506 bytes, checksum: 7f1d951de971fd3b2fac07078a29d3e6 (MD5) / No presente trabalho apresentamos alguns aspectos da Teoria de Álgebras de Funções. Estudamos a dualidade de Gelfand entre espaços topológicos compactos e álgebras C*c. Usando os resultados desse estudo, desenvolvemos a dualidade de Gelfand entre grupos topológicos compactos e álgebras C*c de Hopf, simples com co-identidade.

Chemnitz, Techn. Univ., Diss., 2007.

The spatial numerical range of an operator on a normed linear space and the algebra numerical range of an element of a unital Banach algebra, as developed by G. Lumer and F. F. Bonsall, are considered and the theory of such numerical ranges applied to Banach algebra. The first part of the thesis is largely expository as in it we introduce the basic results on numerical ranges. For an element of a unital Banach algebra, the question of approximating its spectrum by numerical ranges has been considered by F. F. Bonsall and J. Duncan. We give an alternative proof that the convex hull of the spectrum of an element may be approximated by its numerical range defined with respect to equivalent renormings of the algebra. In the particular case of operators on a Hilbert space, this leads to a sharper version of a result by J. P. Williams. An element is hermitian if it has real numerical range. Such an element is characterized in terms of the linear subspace spanned by the unit, the element and its square. This is used to characterize Banach*–algebras in which every self–adjoint element is hermitian. From this an elementary proof that such algebras are B*-algebras in an equivalent norm is given. As indicated by T. W. Palmer, a formula of L. Harris is then used to show that the equivalent renorming is unnecessary, thus giving a simple proof of Palmer's characterization of B*-algebras among Banach algebras. The closure properties of the spatial numerical range are studied. A construction of B. Berberian is extended to normed linear spaces, however because the numerical range need not be convex, the result obtained is weaker than that of Berberian for Hilbert spaces. A Hilbert space or an Lp-space, for p between one and infinity, is seen to be finite dimensional if and only if all the compact operators have closed spatial numerical range. The spatial numerical range of a compact operator, on a Hilbert space or an Lp-space, for p between one and infinity, is shown to contain all the non–zero extreme points of its closure. So, for a compact operator on a Hilbert space the spatial numerical range is closed if and only if it contains the origin, thus answering a question of P. R. Halmos. Operators that attain their numerical radius are also considered. A result of D. Hilbert is extended to a class of Banach spaces. In a Hilbert space the hermitian operators, which attain their numerical radius, are shown to be dense among all the hermitian operators. This leads to a stronger form of a result by J. Lindenstrauss in the spatial case of operators on a Hilbert space. / PhD Doctorate

In this thesis we study Wiener’s lemma. The classical version of the lemma, whose realm is a Banach algebra, asserts that the pointwise inverse of a nonzero function with absolutely convergent Fourier expansion, also possesses an absolutely convergent Fourier expansion. The main purpose of this thesis is to investigate the validity inalgebras endowed with a quasi-norm or a p-norm.As a warmup, we prove the classical version of Wiener’s lemma using elemen-tary analysis. Furthermore, we establish results in Banach algebras concerning spectral theory, maximal ideals and multiplicative linear functionals and present a proof Wiener’s lemma using Banach algebra techniques. Let ν be a submultiplicative weight function satisfying the Gelfand-Raikov-Shilov condition. We show that if a nonzero function f has a ν-weighted absolutely convergent Fourier series in a p-normed algebra A. Then 1/f also has a ν-weightedabsolutely convergent Fourier series in A.

The dissertation contributes to the further advancement of the theory of various classes of discrete and continuous (integral) convolution operators. The thesis is devoted to the study of sequences of matrices or operators which are built up in special ways from generalized discrete or continuous convolution operators. The generating functions depend on three variables and this leads to considerably more complicated approximation sequences. The aim was to obtain for each case a result analogous to the first Szegö limit theorem providing the first order asymptotic formula for the spectra of regular convolutions.

In this thesis, we aim to study derivations from l^1(Z+) to its dual, l^\infty(Z+). We first characterize them as certain closed subspace of l^1(Z+). Then we present a necessary and sufficient condition, due to M. J. Heath, to make a bounded derivation on l^1(Z+) into l^\infty(Z+), a compact linear operator. After that base on the work in [6], we study weakly compact derivations from l^1(Z+) to its dual. We introduce T-sets and TF-sets and then state their relation with weakly compact operators on l^1(Z+). These results are originally due to Y. Choi and M. J. Heath, but we give simpler proofs. Finally, we will study certain classes of derivations from L^1(R+) to L^\infty(R+), and give an elementary proof that they are always mapped into C0(R+).

In this thesis we discuss amenability properties of the Banach algebra-valued continuous functions on a compact Hausdorff space X. Let A be a Banach algebra. The space of A-valued continuous functions on X, denoted by C(X,A), form a new Banach algebra. We show that C(X,A) has a bounded approximate diagonal (i.e. it is amenable) if and only if A has a bounded approximate diagonal. We also show that if A has a compactly central approximate diagonal then C(X,A) has a compact approximate diagonal. We note that, unlike C(X), in general C(X,A) is not a C*-algebra, and is no longer commutative if A is not so. Our method is inspired by a work of M. Abtahi and Y. Zhang. In addition to the above investigation, we directly construct a bounded approximate diagonal for C0(X), the Banach algebra of the closure of compactly supported continuous functions on a locally compact Hausdorff space X.

By Banach algebra we mean Banach space enriched with a multiplication operation. It is a mathematical structure that is used in the non-periodic homogenization of composite materials. The theory of classical homogenization studies materials assuming the periodicity of the structure. For real materials, the assumption of a periodicity is not enough and is replaced by the so-called an abstract hypothesis based on a concept composed mainly of the spectrum of Banach algebra and Sigma convergence. This theory was introduced in 2004.

The dissertation contributes to the further advancement of the theory of various classes of discrete and continuous (integral) convolution operators. The thesis is devoted to the study of sequences of matrices or operators which are built up in special ways from generalized discrete or continuous convolution operators. The generating functions depend on three variables and this leads to considerably more complicated approximation sequences. The aim was to obtain for each case a result analogous to the first Szegö limit theorem providing the first order asymptotic formula for the spectra of regular convolutions.

Viele Probleme der Riss- und Bruchmechanik sowie der mathematischen Physik lassen sich auf Lösungen von singulären Integralgleichungen über einem Intervall zurückführen. Diese Gleichungen setzen sich im Wesentlichen aus dem Cauchy'schen singulären Integraloperator und zusätzlichen Integraloperatoren mit festen Singularitäten in den jeweiligen Kernen zusammen. Zur numerischen Lösung solcher Gleichungen werden polynomiale Kollokations-Quadraturverfahren betrachet. Als Ansatzfunktionen und Kollokationspunkte werden dabei gewichtete Polynome und Tschebyscheff-Knoten gewählt. Die Gewichte sind so gewählt, dass diese das asymptotische Verhalten der Lösung in den Randpunkten widerspiegeln. Mit Hilfe von C*-Algebra Techniken, werden in dieser Arbeit notwendige und hinreichende Bedingungen für die Stabilität der Kollokations-Quadraturverfahren angegeben. Die theoretischen Resultate werden dabei durch numerische Berechnungen anhand des Problems der angerissenen Halbebene und des angerissenen Loches überprüft.