Gerard Buskes

Small Riesz spaces

Many facts in the theory of general Riesz spaces are easily verified by thinking in terms of spaces of functions. A proof via this insight is said to use representation theory. In recent years a growing number of authors has successfully been trying to bypass representation theorems, judging them to be extraneous. (See, for instance, [9,10].) In spite of the positive aspects of these efforts the following can be said. Firstly, avoiding representation theory does not always make the facts transparent. Reading the more cumbersome constructions and procedures inside the Riesz space itself one feels the need for a pictorial representation with functions, and one suspects the author himself of secret heretical thoughts. Secondly, the direct method leads to repeating constructions of the same nature over and over again.

Vector Lattice Powers: f-Algebras and Functional Calculus

Let s ∈ {2.3,…} and E be an Archimedean vector lattice. We prove that there exists a unique pair (E Ⓢ ,Ⓢ), where E Ⓢ is an Archimedean vector lattice and Ⓢ:E× ··· ×E (s times) → E Ⓢ is a symmetric lattice s-morphism, such that for every Archimedean vector lattice F and every symmetric lattice s-morphism T:E × ··· × E (s times) → F, there exists a unique lattice homomorphism T Ⓢ :E Ⓢ → F such that T = T Ⓢ ○Ⓢ. We give two approaches to construct (E Ⓢ ,Ⓢ) based on f-algebras and functional calculus, respectively, provided that E is also uniformly complete.

Bounded variation and tensor products of banach lattices

We introduce bilinear maps of order bounded variation, semivariation and norm bounded variation. We use these notions to extend the knowledge of the projective tensor product of Banach lattices.

Almost f-algebras: Structure and the Dedekind completion

We prove a representation theorem for almost f-algebras, from which we infer the existence of almost f-algebra multiplications on the Dedekind completions of almost f-algebras.

The de Schipper formula and squares of Riesz spaces

Abstract In this paper we introduce and study the square mean and the geometric mean in Riesz spaces. We prove that every geometric mean closed Riesz space is square mean closed and give a counterexample to the converse. We define for positive a, b in a square mean closed Riesz space E an addition via the formula a ⊞ b =sup {(cos x ) a + (sin x ) b : 0 ⩽ x ⩽ 2π}, which goes back to a formula by de Schipper. In case that E is geometric mean closed this turns the mldeflying set of the positive cone of E into a lattice ordered semigroup, which in turn is the positive cone ofa Riesz space E □ . We prove, under the additional condition that E is geometric mean closed, that E □ is Riesz isomorphic to the square of E as introduced earlier by Buskes and van Rooij.

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map ϕ from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that ϕ(f)(y)=ω(y)fh(y)for allf∈C(X)andy∈Y. The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that ϕ(a)=ωψ(a)for alla∈A. We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map ϕ from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.

The Archimedean ℓ-group tensor product

We introduce a construction (inZ F-set theory) for the Archimedean ℓ-group tensor product. We relate this tensor product to the existing ones in the theory of Archimedean vector lattices and ℓ-groups.

Polar decomposition of order bounded disjointness preserving operators

We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded disjointness preserving operator. In this way, we constructively generalize results by Abramovich, Arensen and Kitover (1992), Grobler and Huijsmans (1997), Hart (1985), Kutateladze, and Meyer-Nieberg (1991).

Abstract M- and Abstract L-Spaces of Polynomials on Banach Lattices

In this paper we use the norm of bounded variation to study multilinear operators and polynomials on Banach lattices. As a result, we obtain when all continuous multilinear operators and polynomials on Banach lattices are regular. We also provide new abstract M- and abstract L-spaces of multilinear operators and polynomials and generalize all the results by Grecu and Ryan, from Banach lattices with an unconditional basis to all Banach lattices.

A Note on Bi-orthomorphisms

We show that the space of bi-orthomorphisms forms a vector lattice. The space of orthomorphisms on a semiprime f-algebra is a vector sublattice of the space of bi-orthomorphisms and an ideal in the case that the f-algebra is Dedekind complete.