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

Abstract

We prove that if an F-lattice E is locally bounded (i.e., an open ball in E centered at 0 is topologically bounded) then every orthomorphism in E is central: Or th(E) = Z (E). This solves partially a problem raised recently by Chil and Meyer.For the Nakano (non-Banach) F-lattice E = ℓ ( p n ) , 0 < p n < 1, the above implication becomes an equivalence.We also study the problem of central orthomorphism in the class of (non-Banach) Musielak-Orlicz sequence spaces ℓ Φ . We give a sufficient condition on Φ for a non-central orthomorphism on ℓ Φ to exist.

A relationship between the space of orthomorphisms and the centre of a vector lattice revisited

“…for Archimedean ordered vector spaces, has been introduced by Buck [3], has received a great deal of attention, see [2,4,6,[10][11][12] and has raised many problems. One of the major problem is the question of when Orth(A) = Z (A).…”

“…There are many cases in which it is known whether or not this is possible. There is some results dealing with the subject [4,[10][11][12]. Recently Toumi [10], rephrased the Wickstead problem in terms of non trivial maximal algebra ideals of the f -algebra Orth(A).…”