Rank one and finite rank elements of Banach algebras

[摘要] ENGLISH ABSTRACT: Let A be a unital complex Banach algebra, which we shall simply refer to asa 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 ifthe 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 semiprimeBanach 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 oneelement 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 asthe finite sums of the rank one elements in the preceding definition, togetherwith 0. At about the same time, J.A. Erdos, S. Giotopoulos and M.S. Lambrouintroduced another definition of rank one elements in semi prime Banachalgebras, for which the following is an equivalent formulation: A non-zeroelement u in a semiprime Banach algebra A is rank one if and only ifu is single and acts compactly on A. Since then various other authors havecontributed to the topics of rank one and finite rank elements, yielding severalcharacterizations and another definition of rank one elements: An element uin a semiprime Banach algebra A is a rank one element if #Sp'(xu) :::; 1for all x E A. This led to another definition of a finite rank element: Anelement u in a semiprime Banach algebra A is a finite rank element ifthere 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 threenotions of rank one and the relationship between the two concepts of finiterank. Some consequences of these relationships will be discussed. An applicationof rank one elements to a perturbation result of B. Aupetit will alsobe included.

[发布机构] Stellenbosch University