Generalized domination in ordered Banach algebras: an introduction

Let $A$ be an ordered Banach algebra and $a,b \in A$ such that $0 \leq a \leq b$. The domination problem involves finding natural conditions under which properties of $b$ will be inherited by $a$. In this paper we consider the more general problem where the assumption $0 \leq a \leq b$ is replaced by the weaker condition $\pm a \leq b$.