Domination by ergodic elements in ordered Banach algebras (Q743471)

An element \(a\) of a Banach algebra is called ergodic if the sequence \[ \frac1n\sum_{k=0}^{n-1}a^k \] converges. The authors consider elements \(a\) and \(b\) in an ordered Banach algebra that satisfy the inequality \(0\leq a\leq b\). They study the problem under which conditions the fact that \(b\) is ergodic implies that \(a\) is ergodic.