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arXiv · 2609.33963

Tensor Products of Almost \(f\)-Algebras: A Counterexample

Mohamed Amine Ben Amor
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Abstract

We show that the natural tensor product multiplication of two Archimedean almost \(f\)-algebras does not, in general, extend to their Fremlin tensor product. The two algebras in the counterexample have the same underlying vector lattice \(C[0,1]\). Their multiplications are explicit, positive and nilpotent of index three. The obstruction is the fact that the function \((s,t)\mapsto e^{st}\) does not belong to \(C[0,1]\ftensor C[0,1]\). The same construction gives Banach almost \(f\)-algebras and has consequences for previously published assertions on Riesz and Fremlin projective tensor products of almost \(f\)-algebras.

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Mohamed Amine Ben Amor. 2026-09-27. Tensor Products of Almost \(f\)-Algebras: A Counterexample. https://arxiv.org/abs/2609.33963

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