Tensor products of Archimedean Riesz spaces: A representational approach

In an earlier paper, Buskes and the author pointed out that, given two Archimedean Riesz spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it is relatively simple to construct, from their Ogasawara-Maeda representations, a third Archimedean Riesz space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a bi-injective Riesz bimorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper E times upper F right-arrow upper G"> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>E</mml:mi> <mml:mo>×</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi :E\times F\to G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. They further pointed out that the Riesz subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis upper E times upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>×</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi (E\times F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to the Archimedean Riesz space tensor product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E ModifyingAbove circled-times With bar upper F"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">E\overline {\otimes }F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> constructed by Fremlin. The proof there relies on the properties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E ModifyingAbove circled-times With bar upper F"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">E\overline {\otimes }F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> proved by Fremlin. In this paper we show that this approach actually gives a simple way to construct and establish the properties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E ModifyingAbove circled-times With bar upper F"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">E\overline {\otimes }F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that any choice of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> yield isomorphic objects.