Extension and inversion of extended orthomorphisms on Riesz spaces

Abstract

Let £ be an Archimedean Riesz space and let Orth°°( E) be the /-algebra consisting of all extended orthomorphisms on E, that is, of all order bounded linear operators T: D -> E, with D an order dense ideal in E, such that T(B n D) C B for every band B in E. We give conditions on E and on a Riesz subspace F of E insuring that every T G Orth°c(7 r ) can be extended to some f E Orth°°(E), and we also consider the problem of inversing an extended orthomorphism on its support. The same problems are also studied in the case of a-orthomorphisms, that is, extended orthomorphisms with a super order dense domain. Furthermore, some applications are given.