C. B. Huijsmans

Averaging operators and positive contractive projections

Abstract The main topic of the present paper is a generalization of two theorems about positive contractive projections on C 0 ( X ) (due to Seever and Kelley, resp.) to the more general structure of ƒ-algebras. The theorems in question are the following: if A is an Archimedean semiprime ƒ-algebra with the Stone condition, then every positive contractive projection T in A satisfies the identity T ( a · Tb ) = T ( Ta · Tb ) for all a , b ϵ A (Seever); furthermore, T is averaging if and only if the range of T is a subalgebra (Kelley). Both theorems hold in particular for an Archimedean unital ƒ-algebra. In contrast to the original proofs of Seever and Kelley in the C 0 (X) -case, our proofs are purely algebraic and order theoretical.

The order bidual of lattice ordered algebras

Abstract The main topic of the present paper is a systematic investigation of the second order dual A ″ of an Archimedean ƒ-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 ƒ-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.

Lattice-Ordered Algebras and f-Algebras: A Survey

In this paper we present a survey of results on lattice-ordered algebras, particularly on f-algebras, almost f-algebras, and d-algebras. Example 1.2(v) and the description of nilpotents in various complex lattice- ordered algebras (Section 6) have not appeared before.

Sums of lattice homomorphisms

Let E and F be Riesz spaces and T 1 , T 2 , ..., T n be linear lattice homomorphisms (henceforth called lattice homomorphisms) from E to F. If T=Σ i=1 n T i , then it is easy to check that T is positive and that if x 0 , x 1 ,...,x n ∈E and x i Λx j =0 for all i¬=;j, then Λ i=0 n Tx i =0. The purpose of this note is to show that if F is Dedekind complete, the above necessary condition for T to be the sum of n lattice homomorphisms is also sufficient. The result extends to sums of disjointness preserving operators, thereby leading to a characterization of the ideal of order bounded operators generated by the lattice homomorphisms

Operator Theory in Function Spaces and Banach Lattices

This volume is dedicated to A.C. Zaanen, one of the pioneers of functional analysis, and eminent expert in modern integration theory and the theory of vector lattices, on the occasion of his 80th birthday. The book opens with biographical notes, including Zaanens curriculum vitae and list of publications. It contains a selection of original research papers which cover a broad spectrum of topics about operators and semigroups of operators on Banach lattices, analysis in function spaces and integration theory. Special attention is paid to the spectral theory of operators on Banach lattices; in particular, to the one of positive operators. Classes of integral operators arising in systems theory, optimization and best approximation problems, and evolution equations are also discussed. The book will appeal to a wide range of readers engaged in pure and applied mathematics.

Approximating Derivative Securities in f -Algebras

If E is a finitely generated Riesz subspace of C(X), where X is a compact Hausdorff space, containing the unit e, then Brown-Mertens- Ross have shown that the uniform closure of E is isomorphic to the uniform closure of the tensor product of the Riesz subspaces generated by s i , e (i = 1,…, m), where s 1, s 2,…, s m generate E. We extend their theorem to Archimedean f-algebras with unit and give applications to the theory of financial markets.

On von Neumann regular f-algebras

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

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) σ(T)= {1} implies T = I; and ii) r(T−I) < 1 implies T Є Z(E), the center of E.

Finitely Generated Vector Sublattices

Let E + be the positive cone of an Archimedean vector lattice E. It is shown in [4, Theorem 2.1] that for arbitrary u, v ∈ E + the vector sublattice R(u, v) of E generated by u and v can be described as follows: