Extensions of orthomorphisms

Abstract

We consider, on an Archimedean Riesz space, the spaces of all linear operators lying between two multiples of the identity (for the order), those leaving all ideals invariant and the order bounded orthomorphisms. We find, if E is uniformly complete, necessary and sufficient conditions for all such operators defined on sublattices of E to extend to the whole of E. Examples are given to show the role of uniform completeness. For the space of all orthomorphisms we give a sufficient condition on E for such an extension to exist.

The problem of central orthomorphisms in a class of F-lattices

“…(Notice that, in this case, inequality (14) holds if and only if sup n∈N p n ln n ≤ 2; e.g., for p n = 1 1+ 1 2 ln n , n = 1, 2, . .…”

Polar decomposition of order bounded disjointness preserving operators

“…Alternatively, one can use the Hahn-Banach Theorem 8.15 in [3]. In fact, Wickstead in [14] proved that any orthomorphism in the centre of any ideal of a Riesz space E can be extended to an orthomorphism on all of E if and only if E is Dedekind complete.…”

Extension and inversion of extended orthomorphisms on Riesz spaces

“…Section 2 is mainly devoted to the following problem: given a Riesz subspace F of E and T e Orth°°(i r ), does there exist some f E Orth°°(£') extending T in some sense? The same kind of problem was considered by Wickstead [16] in the case of orthomorphisms; although Orth°°(£) has generally much more elements than Orth(is), the extension of extended orthomorphisms does not require so strong hypothesis as the extension of orthomorphisms does and, in fact, the two problems and their solutions are very different. The concept of a quasi-unital Riesz subspace in an /-algebra, introduced here, seems to be specially useful in that setting.…”