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Topics in Uniform Algebras and Banach Function Algebras by Taher

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Topics in Uniform Algebras and Banach Function Algebras by Taher Topics in Uniform Algebras and Banach Function Algebras by Taher Ghasemi-Honary Department of Mathematics, University for Teacher Education, 49, Roosevelt Avenue, Teheran, Iran. A Thesis submitted for the degree of Ph.D. in the University of London and for the Diploma of Imperial College. Department of Mathematics, Imperial College of Science and Technology, London, S.W.7. June, 1976 2 Abstract In the introductory chapter a brief outline of those parts of commutative Banach algebras and uniform algebras which form a basis for the following chapters is given. In particular the definitions and notations which we shall use throughout the thesis is established. Since the theory of peak points and boundaries of uniform algebras is well-developed, we give details of the known properties of them in the first chapter. The main object of this thesis is to study the relation between peak points and the Shilov boundary of Banach function algebras. In Chapter two the density of peak ooints in the Shilov boundary of a Banach function algebra is discussed and then some examples of Banach function algebras which contain peak sets without peak points are given. We also give some examples of uniform function spaces with certain properties. In Chapter three we consider a particular class of Banach function algebras of infinitely differentiable functions on compact plane sets such that the algebras are quasianalytic. This class was introduced by Dales & Davie [11] who have given an example of a natural Banach function algebra whose set of peak points is countable, and hence is of first category in the Shilov boundary. I have also given such an example on the Swiss cheese for which the set of peak points is uncountable but it is still of first category in the Shilov boundary. The fact that the maximal ideal space of H is not first countable is proved in Chapter four, and it is used to construct Banach function algebras having certain properties. An example of a uniform algebra on the closed unit disk is also given which is not finitely generated. Finally Chapter five contains some propositions concerning the question arising from Gamelin [15]. I have given some conditions on the Banach algebra A with MA = IT such that T = b75- = r(A) or Tc r(A). 3 Acknowledgements I am indebted to my supervisor, Professor J.G. Clunie for suggesting the topic of this thesis and for his guidance during my studies. He specially helped me in constructing the counter-examples. I also offer my thanks to Miss Pindelska of the Mathematics Library, who obtained for me the papers and books which I needed during my studies. Special thanks are due to the University for Teacher Education in Teheran who supported me financially throughout my postgraduate studies. I am most grateful to Mrs. M. Robertson for her care and patience in typing the thesis. 4 Contents Page Chapter one: Introduction 5 1-1 Banach Algebras 5 1-2 Function Algebras 13 1-3 Function Spaces 15 1-4 Uniform Algebras 21 1-5 Peak Points and Boundaries of Uniform Algebras 27 1-6 The Union of Peak Sets 31 Chapter two: Peak Sets and Peak Points of Banach Function Algebras 33 2-1 The existence of the smallest boundary 33 2-2 The density of the peak points in the Shilov boundary 38 2-3 Examples of Banach function algebras having peak sets without peak points 47 2-4 Examples of uniform function spaces 81 Chapter three: Banach Function Algebras on Compact Plane Sets 86 3-1 Definitions and basic properties 86 3-2 Peak points and the Shilov boundary 94 Chapter four: Counter Examples 100 4-1 Preliminaries 100 4-2 Concerning maximal ideal space of H 102 4-3 Examples of uniform algebras having certain properties 106 Chapter five: An Open Question on The Shilov Boundary of Banach Algebras 113 5-1 Preliminaries 113 5-2 Concerning finitely generated uniform algebras 115 5-3 Certain Banach algebras 118 References 121 5 Chapter one Introduction The theory of Banach function algebras and in particular uniform algebras draws from three sources: functional analysis, analytic functions of several complex variables and general topology. The subject of uniform algebras has been receiving an increasing amount of attention in recent years. The theory of peak points and boundaries for uniform algebras is specially well-developed. But the theory of Banach function algebras has not been developed too much. The main object of this thesis is to study the situation which can occur for Banach function algebras. I shall also study the relation between peak points and boundaries of uniform function spaces and Banach function spaces. In this introductory chapter a brief outline of those parts of commutative Banach algebras and uniform algebras which form a basis for the following chapters is given. In particular the definitions and notations which we shall use throughout the thesis is established. Since the theory of peak points and boundaries of uniform algebras is well-developed, we give details of the known properties of them. Some of the results stated in this chapter are only relevant to the other parts and will not be referred to in the remainder of the thesis. References are given in the form [15] or [15; chap. II], which refer to reference 15. 6 1-1 Banach Algebras (1.1.1) Let (A, 11• II) be a commutative normed algebra with unit 1 over the complex field C . Since for every f,g e A, Ilfg11 < 11f11 .11g11 we have 11111 > 1. We suppose that 11111 = 1 and it can be shown that this leads to no loss of generality. From now on by a normed algebra we mean a commutative normed algebra over d with unit 1 such that 11111 = 1. If the normed algebra (A, 11'11) is complete under 11.11' it is called a Banach algebra. Let A be a normed algebra. f e A is invertible if there exists g e A such that f.g = 1. g is called the inverse of f and is denoted -1 1 1 by f or . The family of invertible elements of A is denoted by A . The spectrum of f e A in A is denoted by aA(f) and is given by 1 crA(f) = {X ee -A c A }. If A is a Banach algebra, it is well-known that aA(f) # 0. Let A be a normed algebra and A be the norm completion of A so that A is a Banach algebra. Since °A(f) C aA(f) and aA(f) # 0, we have aA(f) # 0. The following theorem is a direct result of aA(f) #cP. Theorem 1 (Gelfand-Mazur Theorem) Any normed algebra which is a field is isometrically isomorphic to the complex field C . Definition. A non-zero complex homomorphism on A is called a character. Let J be a closed maximal ideal of the normed algebra (A, 11'11). The quotient algebra A/J, in the usual quotient norm, is also a normed algebra over C with the unit 1 + J. Since 111+g11 > 1 for all g e J we have 111+J11 > 1. As 11111 = 1 we conclude that 111+J11 = 1. 7 Moreover every non-zero element of A/J is invertible and so by the Gelfand-Mazur Theorem A/J is isometrically isomorphic to C . Since the projection 4):A A/J is a continuous homomorphism with kernel J, then J is the kernel of a continuous character 4) on A; i.e. J = {f e A: 4)(f) = 01. Conversely, if 4) is a continuous character on A and M is the kernel of 4), then M¢ is a closed maximal ideal of A. Consequently the correspondence 4) -* M(1) is a 1-1 correspondence between the continuous characters on A and the closed maximal ideals of A. Clearly 1¢1I = 4)(1) = 1 for any continuous character 4) on A. It is customary to identify each closed maximal ideal of A with the continuous character that it determines. If A is a Banach algebra, every maximal ideal is closed; equivalently every character on A is continuous. (1.1.2) Let A be a normed algebra. The maximal ideal space (spectrum or carrier space) of A, denoted by MA, is defined as the set of all closed maximal ideals of A; or equivalently, the set of all continuous characters on A. Clearly MA is a subset of the unit ball of the conjugate space A* of A. We define the topology of MA to be the weak*- * topology that MA inherits from A . In other words, a net {4)a} in MA converges to 4) if and only if ¢a (f) 4)(f) for all f 6 A. A basis of open neighbourhoods of sbo 6 MA is given by the sets of the form N(¢0;f1,f2, fri;E) = f4J C MA:14/(fj) 4/0(fi) l < C. 1<j<n/ where e > 0, n is a positive integer and fl,f2, fn a A. Since A* is a Hausdorff space with the weak*-topology, MA is also Hausdorff. By Alaoglu's theorem, the closed unit ball of A* is weak*-compact and 8 so MA, being a weak -closed subset of the closed unit ball, is also weak -compact. (1.1.3) Let A be a normed algebra. The Gelfan4 transform of f e A is the complex-valued function f on MA defined by t(4) = 4(f). Clearly A = {E:f e Al is a normed algebra of continuous functions on MA under the uniform norm = sup 'NO = sup 14)(f) I.
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