Bulletin of the Australian Mathematical Society

Research Article

A spectral problem in ordered Banach algebras

S. Moutona1

a1 Department of Mathematics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa, e-mail: smo@sun.ac.za


We recall the definition and properties of an algebra cone C of a complex unital Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A, and A is then called an ordered Banach algebra. The Banach algebra xs2112(E) of all bounded linear operators on a complex Banach lattice E is an example of an ordered Banach algebra, and an interesting aspect of research in ordered Banach algebras is that of investigating in an ordered Banach algebra-context certain problems that originated in xs2112(E). In this paper we investigate the problems of providing conditions under which (1) a positive element a with spectrum consisting of 1 only will necessarily be greater than or equal to 1, and (2) f (a) will be positive if a is positive, where f (a) is the element defined by the holomorphic functional calculus.

(Received July 16 2002)