Bounded orthomorphisms between locally solid vector lattices

R. Sabbagh, O. Zabeti

University of Sistan and Baluchestan, P.O. Box 98135-674, Zahedan, Iran

Abstract: The main aim of the present note is to consider bounded orthomorphisms between locally solid vector lattices. We establish a version of the remarkable Zannen theorem regarding equivalence between orthomorphisms and the underlying vector lattice for the case of all bounded orthomomorphisms. Furthermore, we investigate topological and ordered structures for these classes of orthomorphisms, as well. In particular, we show that each class of bounded orthomorphisms possesses the Levi or the $AM$-properties if and only if so is the underlying locally solid vector lattice. Moreover, we establish a similar result for the Lebesgue property, as well.

Key words: orthomorphism, bounded orthomorphism, $f$-algebra, locally solid vector lattice.

UDC: 517.98

MSC: 46A40, 47B65, 46A32

Received: 22.12.2020

Language: English

DOI: 10.46698/c1197-8093-8231-u