Domination and generalized domination in ordered Banach algebras
In ordered Banach algebra (OBA) theory, various authors have studied the so-called domination problem (see, e.g., Section 4.2 in [4]): given two elements \(a\) and \(b\) of an OBA such that \(0 \leq a \leq b\), under what hypotheses are properties of \(b\) inherited by \(a\)? In 2014, this problem was studied for ergodicity by Mouton and Muzundu [2], yielding the so-called Ergodic Domination Theorem. The following open questions appeared in the survey paper [4]:
- Can the Ergodic Domination Theorem be extended by replacing the condition \(0 \leq a \leq b\) with the weaker condition \(\pm a \leq b\)?
- Can this theorem be obtained without the so-called “weak monotonicity” assumption?
The aim of this talk is to discuss our (partial) answers to these open questions. After introducing the necessary background, the concept of generalized domination in OBAs shall be introduced. This will be followed by the first main result, which will appear in [3] and shows our investigations to the first problem above. Our approach to the second problem uses Alekhno’s concepts of irreducibility and OBAs with disjunctive products in [1]. Our main result in this regard shall be presented in the last part of the talk.
References
[1] E.A. Alekhno. The irreducibility in ordered Banach algebras. Positivity, 16(1):143–176 (2012).
[2] S. Mouton and K. Muzundu. Domination by ergodic elements in ordered Banach algebras. Positivity, 18(1): 119–130 (2014).
[3] S. Mouton and A.D. Rabearivony. Generalized domination of ergodic elements in ordered Banach algebras. Quaestiones Mathematicae, (to appear).
[4] S. Mouton and H. Raubenheimer. Spectral theory in ordered Banach algebras. Positivity, 21(2): 755–786 (2017).