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Abstract

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.