An alternative proof of a Radon-Nikodym theorem for lattice homomorphisms

We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.     

The order bidual of lattice ordered algebras

The main topic of the present paper is a systematic investigation of the second order dual A″ of an Archimedean ?-algebra A with point separating order dual A′. It is shown that in the case that A has a unit element, the equality A″ = (A′)′_n holds, where (A′)′_n is the collection of all order continuous linear functionals on A′. It turns out that in general (A′)′_n, equipped with the Arens multiplication is an ?-algebra again. Necessary and sufficient conditions are derived for (A′)′_n to be semiprime and for (A′)′_n to have a unit element with respect to this multiplication.     

Some remarks on disjointness preserving operators

In this note we present a simple proof of the following results: if T: E E is a lattice homomorphism on a Banach lattice E, then: i) (T)={1} implies T=I; and ii) r(T–I)<1 implies TZ(E), the center of E.     

The projection onto the center of an ℓ-algebra

In a uniformly complete -algebra with unite>0 the principal band {e} ^dd generated bye is a projection band. We prove that the formula of A. R. Schep for the projection onto this band which holds in the -algebra of order bounded operators on a Dedekind complete vector lattice, in general defines a projection onto the commutant of {e} ^dd . We present some examples to show that the commutant may be strictly larger than the band {e} ^dd , and also conditions which guarantee equality.This paper was written during the fall semester of 1993 while the first author visited the Department of Mathematics of the University of Leiden as a guest of Dr. C. B. Huijsmans, to whom the express his gratitude for the hospitality offered to him.     

Bands in lattices of operators

We consider the lattice of regular operators on a Dedekind complete Banach lattice. We show that in general the projection onto a band generated by a lattice homomorphism need not be continuous and that the principal bands need not be closed for the operator norm. In fact it is possible to find a convergent sequence of operators all the members of which are disjoint from the limit.

     

Disjointness Preserving Operators on Complex Riesz Spaces

It is proven that ifE and F are complex Riesz spaces and ifT is an order bounded disjointness preserving operator fromE intoF , then This fundamental result of M. Meyer is obtained by elementary means using as the main tool the functional calculus derived from the Freudenthal spectral theorem. It is also shown that ifT is an order bounded disjointness preserving operator, a formula of the form holds. It implies a polar decomposition of an order bounded disjointness preserving operator as the product of a Riesz homomorphism and an orthomorphism. Results of P. Meyer-Nieberg in this regard are generalized.     

On von Neumann regular f-algebras

It is shown that for a large class of f-algebras, von Neumann regularity and -lateral completeness are equivalent notions.