Disjointness preserving operators on complex Riesz spaces
For complex Riesz spaces, an order bounded disjointness preserving operator \(T\) satisfies \(|Tz|=|T|z||\) for \(z\) in the domain. This result, due to M. Meyer, is proved here without using representation theory. The proof uses a functional calculus on uniformly complete Riesz spaces. In this setting, each complex Riesz space is viewed as the complexification of a real Riesz space. These techniques are then used to show that \(Tz= \text{sgn }T(|z|)\cdot|T|(z)\). This then leads to decomposition of \(T\) as the product of a Riesz homomorphism and an orthomorphism. This polar decomposition generalizes results of Meyer-Nieberg while relying on only elementary methods for the proof. Conditions implying the existence of the polar decomposition relate to the \(Z\)-extension property for principal ideals; a subspace \(L\) of a Riesz space \(E\) is said to have the \(Z\)-extension property if every orthomorphism in the center \(Z(L)\) of \(L\) has an extension to a central operator in \(Z(E)\). Further, \(\sigma\)-Dedekind completeness implies the \(Z\)-extension property for all principal ideals.
order bounded disjointness preserving operator
product of a Riesz homomorphism and an orthomorphism
scientific article; zbMATH DE number 1121763