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Gerard Buskes

Publications and source records attributed to Gerard Buskes.

8 recordsLinked to original sources
arXiv2023

On the Boolean algebra tensor product via Caratheodory spaces of place functions

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin's Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin's problem 315Y(f) in \cite{fremlin_measure} concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions.

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arXiv2022

Two Results on Fremlin's Archimedean Riesz Space Tensor Product

In this paper, we characterize when, for any infinite cardinal {\alpha}, the Fremlin tensor product of two Archimedean Riesz spaces is Dedekind {\alpha}-complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself.

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arXiv2019

Polynomial Valuations on Vector Lattices

We prove that polynomial valuations on vector lattices correspond to orthosymmetric multilinear maps. As a consequence we obtain a concise proof of the equivalence of orthosymmetry and orthogonal additivity.

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arXiv2018

Characterizing Bounded Orthogonally Additive Polynomials on Vector Lattices

We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in \cite{Kusa} actually characterize them. Secondly, by employing complexifications of the unique symmetric multilinear maps associated with orthogonally additive maps we derive new characterizing formulas.

math.FA↗
arXiv2017

Vector lattices and $f$-algebras: the classical inequalities

We prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean (real or complex) vector lattice, from which the Cauchy-Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean (real or complex) $f$-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove the Hölder inequality for weighted geometric mean closed Archimedean (real or complex) $Φ$-algebras, improving results by Boulabiar and Toumi. As a consequence, the Minkowski inequality for weighted geometric mean closed Archimedean (real or complex) $Φ$-algebras is obtained.

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arXiv2014

Functional Completions of Archimedean Vector Lattices

We study completions of Archimedean vector lattices relative to any nonempty set of positively-homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper.

math.FA↗
arXiv2014

Complex Vector Lattices Via Functional Completions

We show that the Fremlin tensor product $C(X)\bar{\otimes}C(Y)$ is not square mean complete when X and Y are uncountable metrizable compact spaces. This motivates the definition of complexification of Archimedean vector lattices, the Fremlin tensor product of Archimedean complex vector lattices, and a theory of powers of Archimedean complex vector lattices.

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