Building on Alekhno’s framework for irreducibility and Frobenius normal forms in ordered Banach algebras (OBAs) with disjunctive products, we prove equality \(a = b\) when with b irreducible and \(r(a) = r(b)\) is a Riesz point of \(\sigma (b)\) with respect to some inessential ideal (see Section 3). In Section 4 we investigate whether positive elements inherit ergodicity from larger positive elements. Our central result establishes that, under certain natural conditions, every positive element a dominated by a (positive) ergodic element b admits an ergodic spectral block, partially resolving one of the open problems from Mouton, S., Raubenheimer, H.: Spectral theory in ordered Banach algebras. Positivity 21(2), 755–786 (2017). Our proofs are operator-free and depend only on Banach algebra techniques, and our results apply to OBAs of operators on Banach lattices with order continuous norms instead of just the regular operators. By using Alekhno’s irreducibility and disjunctive product techniques, we bypass the weak monotonicity assumption which was crucial in Mouton, S., Muzundu, K.: Domination by ergodic elements in ordered Banach algebras. Positivity 18(1), 119–130 (2014).
