On domination of inessential elements in ordered Banach algebras

Abstract

If $A$ is an ordered Banach algebra ordered by an algebra cone $C$, then we reference the following problem as the `domination problem': If $0\leq a\leq b$ and $b$ has a certain property, then does $a$ inherit this property? We extend the analysis of this problem in the setting of radical elements and introduce it for inessential, rank one and finite elements. We also introduce the class of $r$-inessential operators on Banach lattices and prove that if $S$ and $T$ are operators on a Banach lattice $E$ such that $0\leq S\leq T$ and $T$ is $r$-inessential then $S$ is also $r$-inessential.