Rank one and finite rank elements of Banach algebras

Muzundu, Kelvin (2007-12)

Thesis (MSc) -- University of Stellenbosch, 2007.

Thesis

ENGLISH ABSTRACT: Let A be a unital complex Banach algebra, which we shall simply refer to as a Banach algebra. An element u in A is single if xuy = 0, where x, y E A, implies that xu = 0 or uy = 0. We say that u acts compactly on A if the operator x H uxu is compact. For an element x E A the set Sp(x) = { ,\ E C : ,\ - x is not invertible in A} is called the spectrum of x in A. The notation #Sp(x) indicates the number of points in Sp(x) and #Sp'(x) denotes the number of non-zero points in Sp(x). In 1978 J. Puhl introduced and studied rank one elements of semiprime Banach algebras. He gave the following definition for a rank one element: A non-zero element u in a semiprime Banach algebra A is a rank one element if there exists a linear functional fu on A such that uxu = fu(x)u for all x E A. In the same paper he defined finite rank elements as the finite sums of the rank one elements in the preceding definition, together with 0. At about the same time, J.A. Erdos, S. Giotopoulos and M.S. Lambrou introduced another definition of rank one elements in semi prime Banach algebras, for which the following is an equivalent formulation: A non-zero element u in a semiprime Banach algebra A is rank one if and only if u is single and acts compactly on A. Since then various other authors have contributed to the topics of rank one and finite rank elements, yielding several characterizations and another definition of rank one elements: An element u in a semiprime Banach algebra A is a rank one element if #Sp'(xu) :::; 1 for all x E A. This led to another definition of a finite rank element: An element u in a semiprime Banach algebra A is a finite rank element if there exists a positive integer n such that #Sp' ( xu) :::; n for all x E A. The purpose of this thesis is to study the relationship among the three notions of rank one and the relationship between the two concepts of finite rank. Some consequences of these relationships will be discussed. An application of rank one elements to a perturbation result of B. Aupetit will also be included.

AFRIKAANSE OPSOMMING: Laat A 'n komplekse unitere Banach algebra wees, waarna ons bloot as 'n Banach algebra sal verwys. 'n Element u E A is enkel as xuy = 0, waar x, y E A, impliseer <lat XU = 0 of uy = 0. Ons se <lat u kompak op A inwerk as die operator x H uxu kompak is. Vir 'n element x E A word die versameling Sp( x) = {A E C : A - x nie inverteerbaar in A is nie} die spektrum van x in A genoem. Die notasie #Sp(x) dui die aantal punte in Sp(x) aan en #Sp'(x) dui die aantal nienul punte in Sp(x) aan. In 1978 het J. Puhl die rang-een-elemente van semipriem Banach algebras gedefinieer en bestudeer. Hy het die volgende definisie vir 'n rang-een-element gegee: 'n Nienul element u in 'n semipriem Banach algebra A is 'n rangeen-element as daar 'n lineere funksionaal fu op A bestaan sodat uxu = fu(x)u vir alle x E A. In dieselfde artikel het hy eindige-rang-elemente as die eindige somme van die rang-een-elemente in die voorafgaande definisie, tesame met 0 gedefinieer. Op min of meer dieselfde tyd het J.A. Erdos, S. Giotopoulos en M.S. Lambrou 'n antler definisie van rang-een-elemente in semipriem Banach algebras geskep, waarvan die volgende 'n ekwivalente formulering is: 'n Nienul element u in 'n semipriem Banach algebra A is 'n rang-een-element as en slegs as u enkel is en kompak op A inwerk. Sedertdien het 'n aantal antler outeurs bygedra tot die onderwerpe van rang-een- en eindige-rang-elemente, wat verskeie karakteriserings tot gevolg gehad het, asook nog 'n definisie van rang-een-elemente: 'n Element u in 'n semipriem Banach algebra A is 'n rang-een-element as #Sp'(xu) ~ 1 vir alle x EA. Dit het tot nog 'n definisie van 'n eindige-rang-element gelei: 'n Element u in 'n semipriem Banach algebra A is 'n eindige-rang-element as daar 'n positiewe heelgetal n bestaan sodat #Sp' ( xu) ~ n vir alle x E A. Die doel van hierdie tesis is om die verwantskap tussen die drie konsepte van rang een en die verwantskap tussen die twee konsepte van eindige rang te bestudeer. Sommige gevolge van hierdie verwantskappe sal bespreek word. Verder sal 'n toepassing van rang-een-elemente op 'n versteuringsresultaat van B. Aupetit ingesluit word.

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