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Commutativity of the Exponential Spectrum Commutativity of the Exponential Spectrum Aram Gevorgyan Maîtrise en mathématiques Québec, Canada © Aram Gevorgyan, 2015 Résumé Pour l’algèbre de Banach complexe A ayant l’élément unité, on dénote par G(A) l’ensemble des éléments inversibles de A, et par G1(A) on dénote la composante qui contient l’unité. Le spectre de a 2 A est l’ensemble de tous les nombres complexes λ tels que λ1 − a2 = G(A), et le spectre exponentiel de a est l’ensemble de tous les nombres complexes tels que λ1−a2 = G1(A). Évidemment, pour chaque élément de l’algèbre, son spectre exponentiel contient le spectre habituel. Il est bien connu que le spectre habituel a une propriété que l’on nommera «propriété de commutativité». Cela signifie que, pour chaque choix des deux éléments a; b 2 A, nous avons Sp(ab) n f0g = Sp(ba) n f0g, où Sp est le spectre. Avons-nous la même propriété pour les spectres exponentiels? Cette question n’est toujours pas résolue. L’objectif de ce mémoire est d’étudier le spectre exponentiel, et plus particulièrement sa propriété de commutativité. Dans le premier chapitre, nous donnerons les définitions d’algèbre de Banach complexe, spectre et spectre exponentiel de ses éléments, et leurs propriétés de base. Aussi nous établirons des relations topologiques entre les spectres exponentiel et habituel. Dans le deuxième chapitre, nous définirons les fonctions holomorphes sur une algèbre de Banach, et discuterons du problème de la propriété de commutativité de spectre exponentiel, en établissant des résultats positifs connus. Dans le troisième et dernier chapitre, nous examinerons quelques exemples d’algèbres de Ba- nach, décrivant les ensembles G(A) et G1(A), et discuterons de la propriété de commutativité pour ces algèbres. iii Abstract For a complex Banach algebra A with unit element, we denote by G(A) the set of invertible elements of A, and by G1(A) we denote the component of G(A) which contains the unit. The spectrum of a 2 A is the set of all complex numbers λ such that λ1−a2 = G(A), and the exponential spectrum of a is the set of all complex numbers λ such that λ1−a2 = G1(A). Of course for each element of the algebra its exponential spectrum contains the usual spectrum. It is well known that the usual spectrum has the so-called commutativity property. This means that, for any two elements a and b of A, we have Sp(ab) n f0g = Sp(ba) n f0g, where Sp denotes the spectrum. Does this property hold for exponential spectra? This is still an open question. The purpose of this memoir is to study the exponential spectrum, and particularly its commutativity property. In chapter one, we will give definitions of a complex Banach algebra, the spectrum and expo- nential spectrum of its elements, and their basic properties. Also we will establish topological relations between exponential and usual spectra. In chapter two, we will define holomorphic functions on a Banach algebra, and also discuss the commutativity property problem for the exponential spectrum, establishing some known positive results. In the last chapter, we will consider some examples of Banach algebras, describing the sets G(A) and G1(A), and discuss the commutativity property for these algebras. v Contents Résumé iii Abstract v Contents vii Acknowledgments xiii 1 Banach Algebras, Spectrum and Exponential Spectrum1 1.1 Banach Algebras, Definitions and Examples.................... 1 1.2 Invertible Elements and Spectrum ......................... 4 1.3 Holomorphic Functional Calculus.......................... 9 1.4 Exponential Spectrum................................ 13 2 Commutativity of Exponential Spectrum: Positive Results 17 2.1 Special Case, When at Least One of the Two Elements is a Limit of Invertible Elements ....................................... 17 2.2 Other Special Cases ................................. 19 3 Examples of Exponential Spectra 23 3.1 The Wiener Algebra ................................. 23 3.2 The Algebra of Bounded Linear Operators on Infinite-Dimensional Hilbert Space 27 3.3 The Calkin Algebra.................................. 29 3.4 Banach Algebra, With a Disconnected Group of Invertible Elements . 33 Conclusion 37 Bibliography 39 vii To my family ix God used beautiful mathematics in creating the world. Paul Dirac (1902-1984) xi Acknowledgments Several people have been supporting me during my studies and my integration into the social life of Quebec City, since the very first day of my arrival in this town. They were always there to help me with any advice and services I asked for. Without them, my education in Laval university simply couldn’t have progressed to the final stage. First of all I would like to thank my supervisor Professor Thomas Joseph Ransford. Professor Ransford helped me with the very first steps, personally introducing me the Laval University and its personnel who I needed to know during my education. The next and the most valuable help of him was the professional support, scientific discussions, advice he provided through per- sonal meetings. I’m deeply grateful to Professor Ransford, his valuable support was essential at every stage of my education. I also would like to thank all the staff and professors of the Department of Mathematics and Statistics with whom I collaborated or asked for help. They were always kind and ready to help me with any service I needed. Specially I would like to mention Sylvie Drolet and Robert Guénette, whose help and support were very significant for me. I am grateful to my friends in Quebec City, for their advice and support in daily life that was especially valuable at the starting period of my stay in Quebec. Specially I want to thank my friend Anush Stepanyan who advised me to start my education in Laval University and helped me very much with everything I needed. Finally, and most of all, I’m deeply grateful to my family and my friends (in Armenia) who encouraged me and provided all the necessary help to make the decision to move to Canada and to start my education in Laval University. Aram Gevorgyan xiii Chapter 1 Banach Algebras, Spectrum and Exponential Spectrum 1.1 Banach Algebras, Definitions and Examples The material of this chapter is mainly from the book of B. Aupetit Aupetit[1991]. Definition 1.1.1. An algebra is a vector space A over the field C, with a multiplication satisfying the following properties: x(yz) = (xy)z; (x + y)z = xz + yz; x(y + z) = xy + xz; λ(xy) = (λx)y = x(λy) for all x; y; z 2 A, and λ 2 C. If moreover A is a Banach space for a norm k · k and satisfies the norm inequality kxyk ≤ kxkkyk; for all x; y 2 A, we say that A is a complex Banach algebra. In the definition, C could be replaced by any field K, in particular by R, but in this memoir we will deal mainly with complex Banach algebras. A Banach algebra A is called commutative, if for all a; b 2 A we have ab = ba, and with unit, if there is an element 1 2 A with k1k = 1, such that for all a 2 A we have 1a = a1 = a. Obviously if the algebra has a unit, then the unit is unique. Indeed, if there were another e 2 A, such that ea = ae = a for all a 2 A, then e = e1 = 1e = 1. Let us give some examples of Banach algebras. 1 Remark 1.1.2. If A is a Banach algebra without unit, it is always possible to embed it isomet- rically in the Banach algebra with unit A+ = A × C, where the operations and the norm are defined by (x; α) + (y; β) = (x + y; α + β); λ(x; α) = (λx, λα); : (x; α)(y; β) = (xy + βx + αy; αβ); k(x; α)k = kxk + jαj Here e = (0; 1) will be the unit element of A+, and the function x 7! (x; 0) is an isomorphism from A into A+. Example 1.1.3. Let K be a compact set. Then C(K), the space of all complex continuous functions on K, with the supremum norm, is a Banach algebra with unit. If K has only n n elements then C(K) = C with coordinatewise multiplication, and of course C is the simplest commutative Banach algebra. If K is a locally compact space, then the Banach space of all complex continuous functions on K which go to zero at infinity, that is, continuous functions f such that fx : x 2 K; jf(x)j ≥ g is compact for all > 0, with the supremum norm, is a Banach algebra, which has no unit if K is not compact. Example 1.1.4. If K is compact, every closed subalgebra of C(K) is a Banach algebra. For instance if K is a compact subset of C, the following examples of Banach algebras are interesting: The algebra P (K), of continuous functions on K which are uniform limits of polynomials on K; the algebra R(K), of continuous functions on K which are uniform limits of rational functions with poles outside of K; and the algebra A(K), of continuous functions on K which are holomorphic on the interior of K. We have the inclusions P (K) ⊂ R(K) ⊂ A(K) ⊂ C(K), and there are examples of K for which all these inclusions are strict. Example 1.1.5. Let G be a locally compact commutative group and let µ be its Haar measure. Then L1(G) is a Banach algebra if we define multiplication by convolution Z (f ∗ g)(x) = f(xy−1)g(y)dµ(y) G 1 R and the norm by the L -norm kfk1 = G jf(x)jdµ(x).
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