Dissertationen zum Thema „Banach algebras“

Chapter 1 defines the notion of a unitary Banach algebra, and gives various examples. The inheritance of the unitary property of quotients and subalgebras is investigated, the main result being that the class of unitary Banach algebras is exactly the class of quotients of discrete group algebras. One problem that is discussed is whether a unitary subalgebra needs to inherit the unit element. Chapter 2 gives several other characterisations of unitary Banach algebras among norm-unital Banach algebras, in particular by conditions on the numerical range. The topological properties of the unitary Banach algebra are also discussed. Chapter 3 deals with isometric isomorphisms of unitary Banach algebras. In particular it is shown that, for groups G₁ and G₂, and A a norm-unital Banach algebra with connected unitary group, or a unital C^*-algebra, the existence of an isometric isomorphism from l¹ (G₁, A) onto l^l (G₂, A) implies that G₁ and G₂ are isomorphic. If A is commutative then these two results can be generalised to be case of locally compact abelian groups G₁ and G₂, and the Banach algebras L¹(G₁, A) and L¹ (G₂, A).

This thesis is devoted to the study of various properties of Banach function algebras. We are particularly interested in the study of antisymmetric decompositions for uniform algebras and regularity of Banach function algebras. We are also interested in the study of Swiss cheese sets, essential uniform algebras and characterisations of C(X) among its subalgebras. The maximal antisymmetric decomposition for uniform algebras is a generalisation of the celebrated Stone-Weierstrass theorem and it is a powerful tool in the study of uniform algebras. However, in the literature, not much attention has been paid to the study of closed antisymmetric subsets. In Section 1.7 we give a characterisation of all the closed antisymmetric subsets for the disc algebra on the unit circle, and we use this characterisation to give a new proof of Wermer’s maximality theorem. Then in Section 4.1 we give characterisations of all the closed antisymmetric subsets for normal uniform algebras on the unit interval or the unit circle. The two types of regularity points, the R-point and the point of regularity, are important concepts in the study of regularity of Banach function algebras. In Section 3.2 we construct two examples of compact plane sets X, such that R(X) has either one R-point while having no points of regularity, or R(X) has one point of continuity while having no R-points. There are the first known examples of natural uniform algebras in the literature which show that R-points and points of continuity can be different. We then use properties of regularity points to study R(X) which is not regular while having no non-trivial Jensen measures. We also use properties of regularity points in Section 4.2 to study small exceptional sets for uniform algebras. In Chapter 2 we study Swiss cheese sets. Our approach is to regard Swiss cheese sets “abstractly”: we study the family of sequences of pairs of numbers, where the numbers represent the centre and radius of discs in the complex plane. We then give a natural topology on the space of abstract Swiss cheeses and give topological proofs of various classicalisation theorems. It is standard that the study of general uniform algebras can be reduced to the study of essential uniform algebras. In Chapter 5 we study methods to construct essential uniform algebras. In particular, we continue to study the method introduced in [26] to show that some more properties are inherited by the constructed essential uniform algebra from the original one. We note that the material in Chapter 2 is joint work with J. Feinstein and S. Morley and is published in [28, 27]. The material in Chapter 3 is joint work with J. Feinstein and is published in [32]. Section 4.2 contains joint work with J. Feinstein.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.
ENGLISH ABSTRACT: This thesis is a study of a generalisation, due to R. Harte (see [9]), of Fredholm theory in the context of bounded linear operators on Banach spaces to a theory in a Banach algebra setting. A bounded linear operator T on a Banach space X is Fredholm if it has closed range and the dimension of its kernel as well as the dimension of the quotient space X/T(X) are finite. The index of a Fredholm operator is the integer dim T−1(0)−dimX/T(X). Weyl operators are those Fredholm operators of which the index is zero. Browder operators are Fredholm operators with finite ascent and descent. Harte’s generalisation is motivated by Atkinson’s theorem, according to which a bounded linear operator on a Banach space is Fredholm if and only if its coset is invertible in the Banach algebra L(X) /K(X), where L(X) is the Banach algebra of bounded linear operators on X and K(X) the two-sided ideal of compact linear operators in L(X). By Harte’s definition, an element a of a Banach algebra A is Fredholm relative to a Banach algebra homomorphism T : A ! B if Ta is invertible in B. Furthermore, an element of the form a + b where a is invertible in A and b is in the kernel of T is called Weyl relative to T and if ab = ba as well, the element is called Browder. Harte consequently introduced spectra corresponding to the sets of Fredholm, Weyl and Browder elements, respectively. He obtained several interesting inclusion results of these sets and their spectra as well as some spectral mapping and inclusion results. We also convey a related result due to Harte which was obtained by using the exponential spectrum. We show what H. du T. Mouton and H. Raubenheimer found when they considered two homomorphisms. They also introduced Ruston and almost Ruston elements which led to an interesting result related to work by B. Aupetit. Finally, we introduce the notions of upper and lower semi-regularities – concepts due to V. M¨uller. M¨uller obtained spectral inclusion results for spectra corresponding to upper and lower semi-regularities. We could use them to recover certain spectral mapping and inclusion results obtained earlier in the thesis, and some could even be improved.
AFRIKAANSE OPSOMMING: Hierdie tesis is ‘n studie van ’n veralgemening deur R. Harte (sien [9]) van Fredholm-teorie in die konteks van begrensde lineˆere operatore op Banachruimtes tot ’n teorie in die konteks van Banach-algebras. ’n Begrensde lineˆere operator T op ’n Banach-ruimte X is Fredholm as sy waardeversameling geslote is en die dimensie van sy kern, sowel as di´e van die kwosi¨entruimte X/T(X), eindig is. Die indeks van ’n Fredholm-operator is die heelgetal dim T−1(0) − dimX/T(X). Weyl-operatore is daardie Fredholm-operatore waarvan die indeks gelyk is aan nul. Fredholm-operatore met eindige styging en daling word Browder-operatore genoem. Harte se veralgemening is gemotiveer deur Atkinson se stelling, waarvolgens ’n begrensde lineˆere operator op ’n Banach-ruimte Fredholm is as en slegs as sy neweklas inverteerbaar is in die Banach-algebra L(X) /K(X), waar L(X) die Banach-algebra van begrensde lineˆere operatore op X is en K(X) die twee-sydige ideaal van kompakte lineˆere operatore in L(X) is. Volgens Harte se definisie is ’n element a van ’n Banach-algebra A Fredholm relatief tot ’n Banach-algebrahomomorfisme T : A ! B as Ta inverteerbaar is in B. Verder word ’n Weyl-element relatief tot ’n Banach-algebrahomomorfisme T : A ! B gedefinieer as ’n element met die vorm a + b, waar a inverteerbaar in A is en b in die kern van T is. As ab = ba met a en b soos in die definisie van ’n Weyl-element, dan word die element Browder relatief tot T genoem. Harte het vervolgens spektra gedefinieer in ooreenstemming met die versamelings van Fredholm-, Weylen Browder-elemente, onderskeidelik. Hy het heelparty interessante resultate met betrekking tot insluitings van die verskillende versamelings en hulle spektra verkry, asook ’n paar spektrale-afbeeldingsresultate en spektraleinsluitingsresultate. Ons dra ook ’n verwante resultaat te danke aan Harte oor, wat verkry is deur van die eksponensi¨ele-spektrum gebruik te maak. Ons wys wat H. du T. Mouton en H. Raubenheimer verkry het deur twee homomorfismes gelyktydig te beskou. Hulle het ook Ruston- en byna Rustonelemente gedefinieer, wat tot ’n interessante resultaat, verwant aan werk van B. Aupetit, gelei het. Ten slotte stel ons nog twee begrippe bekend, naamlik ’n onder-semi-regulariteit en ’n bo-semi-regulariteit – konsepte te danke aan V. M¨uller. M¨uller het spektrale-insluitingsresultate verkry vir spektra wat ooreenstem met bo- en onder-semi-regulariteite. Ons kon dit gebruik om sekere spektrale-afbeeldingsresultate en spektrale-insluitingsresultate wat vroe¨er in hierdie tesis verkry is, te herwin, en sommige kon selfs verbeter word.

Orientador : Jorge Tulio Mujica Ascui
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Mestrado
Mestre em Matemática

Orientador: Jorge Tulio Mujica Ascui
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação
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Resumo: O principal objetivo deste trabalho é o estudo de certos espaços de Banach de funções analíticas no disco aberto unitário, conhecidos como espaços de Hardy. Um outro objetivo é o estudo das propriedades básicas de álgebras de Banach, com especial ênfase na álgebra do disco e na álgebra das funções analíticas e limitadas no disco aberto unitário
Abstract: The main objective of this work is the study of certain Banach spaces of analytic functions on the open unit disc, known as Hardy spaces. Another objective is the study of the basic properties of Banach algebras, with special emphasis in the disc algebra and the algebra of bounded analytic functions in the open unit disc
Mestrado
Matematica
Mestre em Matemática

This thesis concerns the theory of Banach algebras, particularly those coming from abstract harmonic analysis. The focus for much of the thesis is the theory of the ideals of these algebras. In the final chapter we use semigroup algebras to solve an open probelm in the theory of C*-algebras. Throughout the thesis we are interested in the interplay between abstract algebra and analysis. Chapters 2, 4, and 5 are closely based upon the articles [88], [89], and [56], respectively. In Chapter 2 we study (algebraic) finite-generation of closed left ideals in Banach algebras. Let G be a locally compact group. We prove that the augmentation ideal in L 1 pGq is finitely-generated as a left ideal if and only if G is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use the theory that we have developed to construct some examples of commutative Banach algebras that relate to a theorem of Gleason. In Chapter 3 we turn our attention to topological finite-generation of closed left ideals in Banach algebras. We define a Banach algebra to be topologically left Noetherian if every closed left ideal is topologically finitely-generated, and we seek infinitedimensional examples of such algebras. We show that, given a compact group G, the group algebra L 1 pGq is topologically left Noetherian if and only if G is metrisable. For a Banach space E satisying a certain condition we show that the Banach algebra of approximable operators ApEq is topologically left Noetherian if and only if E 1 is separable, whereas it is topologically right Noetherian if and only if E is separable. We also define what it means for a dual Banach algebra to be weak*-topologically left Noetherian, and give examples which satisfy and fail this condition. Along the way, we give classifications of the weak*-closed left ideals in MpGq, for G a compact group, and in BpEq, for E a reflexive Banach space with AP. Chapter 4 looks at the Jacobson radical of the bidual of a Banach algebra. We prove that the bidual of a Beurling algebra on Z, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that rad p` 1 p‘8 i“1Zq 2 q contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight ω on Z such that the bidual of ` 1 pZ, ωq contains a radical element which is not nilpotent. In Chapter 5 we move away from the theory of ideals and consider a question about the notion of finiteness in C*-alegebras. We construct a unital pre-C*-algebra A0 which is stably finite, in the sense that every left invertible square matrix over A0 is right invertible, while the C*-completion of A0 contains a non-unitary isometry, and so it is infinite. This answers a question of Choi. The construction is based on semigroup algebras.

Thesis (Ph. D.)--University of Oregon, 2002.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 57-58). Also available for download via the World Wide Web; free to University of Oregon users. Address: http://wwwlib.umi.com/cr/uoregon/fullcit?p3055703.

This thesis is a collection of theorems which say something about the following question: if we know that a bounded operator on a commutative unital Banach algebra is a unital endomorphism, what can we say about its other properties? More specifically, the majority of results say something about how the spectrum of a commutative unital Banach algebra endomorphism is dependent upon the properties of the algebra on which it acts. The main result of the thesis (the subject of Chapter 3) reveals that primary ideals (that is, ideals with single point hulls) can sometimes be particularly important in questions of this type. The thesis also contains some contributions to the Fredholm theory for bounded operators on an arbitrary complex Banach space. The second major result of the thesis is in this direction, and concerns the relationship between the essential spectrum of a bounded operator on a Banach space and those of its restrictions and quotients - `to' and `by' - closed invariant subspaces.

Thesis (MSc)--University of Stellenbosch, 2003.
ENGLISH ABSTRACT: Let A and B be unital complex Banach algebras with identities 1 and I' respectively. A linear map T : A -+ B is invertibility preserving if Tx is invertible in B for every invertible x E A. We say that T is unital if Tl = I'. IfTx2 = (TX)2 for all x E A, we call T a Jordan homomorphism. We examine an unsolved problem posed by 1. Kaplansky: Let A and B be unital complex Banach algebras and T : A -+ B a unital invertibility preserving linear map. What conditions on A, Band T imply that T is a Jordan homomorphism? Partial motivation for this problem are the Gleason-Kahane-Zelazko Theorem (1968) and a result of Marcus and Purves (1959), these also being special instances of the problem. We will also look at other special cases answering Kaplansky's problem, the most important being the result stating that if A is a von Neumann algebra, B a semi-simple Banach algebra and T : A -+ B a unital bijective invertibility preserving linear map, then T is a Jordan homomorphism (B. Aupetit, 2000). For a unital complex Banach algebra A, we denote the spectrum of x E A by Sp (x, A). Let a(x, A) denote the union of Sp (x, A) and the bounded components of AFRIKAANSE OPSOMMING: Gestel A en B is unitale komplekse Banach algebras met identiteite 1 en I' onderskeidelik. 'n Lineêre afbeelding T : A -+ B is omkeerbaar-behoudend as Tx omkeerbaar in B is vir elke omkeerbare element x E A. Ons sê dat T unitaal is as Tl = I'. As Tx2 = (TX)2 vir alle x E A, dan noem ons T 'n Jordan homomorfisme. Ons ondersoek 'n onopgeloste probleem wat deur I. Kaplansky voorgestel is: Gestel A en B is unitale komplekse Banach algebras en T : A -+ B is 'n unitale omkeerbaar-behoudende lineêre afbeelding. Watter voorwaardes op A, B en T impliseer dat T 'n Jordan homomorfisme is? Gedeeltelike motivering vir hierdie probleem is die Gleason-Kahane-Zelazko Stelling (1968) en 'n resultaat van Marcus en Purves (1959), wat terselfdertyd ook spesiale gevalle van die probleem is. Ons salook na ander spesiale gevalle kyk wat antwoorde lewer op Kaplansky se probleem. Die belangrikste van hierdie resultate sê dat as A 'n von Neumann algebra is, B 'n semi-eenvoudige Banach algebra is en T : A -+ B 'n unitale omkeerbaar-behoudende bijektiewe lineêre afbeelding is, dan is T 'n Jordan homomorfisme (B. Aupetit, 2000). Vir 'n unitale komplekse Banach algebra A, dui ons die spektrum van x E A aan met Sp (x, A). Laat cr(x, A) die vereniging van Sp (x, A) en die begrensde komponente van