Domination properties in ordered Banach algebras (Q2773410)

Starting from a real or complex Banach algebra \(A\) with unity, \(C\) is a cone of \(A\) ifNEWLINENEWLINENEWLINE1) \(C+C\leq C\)NEWLINENEWLINENEWLINE2) \(\lambda C\leq C\) for all \(\lambda\geq 0\).NEWLINENEWLINENEWLINE\(C\) induces on \(A\) an ordering which is compatible with the algebraic structure of \(A\). \(A\), with this order, is called an ordered Banach algebra; for such an \(A\), the authors consider the following problem:NEWLINENEWLINENEWLINEunder which conditions does it follow from \(0\leq a\leq b\) in \(A\) and `\(b\)' being in the radical of \(A\) that `\(a\)' is in the radical at \(A\)?NEWLINENEWLINENEWLINESome interesting answers are obtained by the use of subharmonic analysis.