SAMS Subject Classification Number: 11

In ordered Banach algebra (OBA) theory, various authors have studied the so-called domination problem (see, e.g., Section 4.2 in [2]): given two elements \(a\) and \(b\) of an OBA such that \(0\le a\le b\), under what hypotheses are properties of \(b\) inherited by \(a\)? We tackle the corresponding problem where the condition \(0\le a\le b\) is replaced by the weaker condition \(\pm a\le b\). (For operators \(S\) and \(T\) on a Banach lattice \(E\) the condition \(\pm S \le T\) means that \(|Sx|\le T|x|\) for all \(x\) in \(E\).) We refer to this as the generalized domination problem. Furthermore, it is presented as an open question in [2] whether the (known) ergodic domination theorem (see [1]) can be extended to this setting. We will show that this question has a positive answer, not only for ergodic domination, but also for most of the existing domination results, including those related to Riesz elements, inessential elements and elements of the radical.

[1] S. Mouton and K. Muzundu, Domination by ergodic elements in ordered Banach algebras, , 18(1)(2014), 119–130.

[2] S. Mouton and H. Raubenheimer,  Spectral theory in ordered Banach algebras, Positivity 21(2) (2017), 755 –786.