# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Domination properties in ordered Banach algebras

### Tom 149 / 2002

Studia Mathematica 149(2002), 63-73 MSC: 46H05, 47A10, 47B65, 06F25. DOI: 10.4064/sm149-1-4

#### Streszczenie

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.

#### Autorzy

• H. du T. MoutonDepartment of Electrical and Electronic Engineering
University of Stellenbosch
Private Bag X1
Matieland 7602, South Africa
e-mail
• S. MoutonDepartment of Mathematics
University of Stellenbosch
Private Bag X1
Matieland 7602, South Africa
e-mail

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