Domination properties in ordered Banach algebras

H. du T. Mouton¹ ; S. Mouton²

¹ Department of Electrical and Electronic Engineering University of Stellenbosch Private Bag X1 Matieland 7602, South Africa
² Department of Mathematics University of Stellenbosch Private Bag X1 Matieland 7602, South Africa

Studia Mathematica, Tome 149 (2002) no. 1, pp. 63-73

Résumé

We recall from [9] the definition and properties of an algebra cone $C$ of a real or complex Banach algebra $A$. It can be shown that $C$ induces on $A$ an ordering which is compatible with the algebraic structure of $A$. The Banach algebra $A$ is then called an ordered Banach algebra. An important property that the algebra cone $C$ may have is that of normality. If $C$ is normal, then the order structure and the topology of $A$ are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties. If $A$ is an ordered Banach algebra with a normal algebra cone $C$, then an important problem is that of providing conditions under which certain spectral properties of a positive element $b$ will be inherited by positive elements dominated by $b$. We are particularly interested in the property of $b$ being an element of the radical of $A$. Some interesting answers can be obtained by the use of subharmonic analysis and Cartan's theorem.

Détail

DOI : 10.4064/sm149-1-4

Keywords: recall definition properties algebra cone real complex banach algebra shown induces ordering which compatible algebraic structure banach algebra called ordered banach algebra important property algebra cone may have normality normal order structure topology reconciled certain ordered banach algebras have interesting spectral properties ordered banach algebra normal algebra cone important problem providing conditions under which certain spectral properties positive element inherited positive elements dominated particularly interested property being element radical interesting answers obtained subharmonic analysis cartans theorem