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arXiv · 2610.13734

The tensor product of the Fremlin tensor product of Banach lattice algebras

Jamel Jaber·Marwa Troudi
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Abstract

We study the Arens regularity of the Fremlin projective tensor product of Banach lattice algebras. We firrst introduce and investigate L-weakly almost periodic functionals, a class of functionals defind in terms of L-weak compactness of their orbits. We obtain a characterization of these functionals in terms of bounded disjoint sequences and use it to analyze the Fremlin projective tensor product. In particular, we prove that, for nontrivial Banach attice algebras A and B, Arens regularity of A j jB implies Arens regularity of both A and B. The converse, however, fails: we construct two Arens regular Banach lattice algebras whose Fremlin projective tensor product is not Arens regular. We also establish several sufficient conditions for the Arens regularity of A j jB, including conditions involving the positive Schur property and order continuity of the norm on the space of regular operators. Finally, we show that when both factors are Banach f-algebras, their Fremlin projective tensor product is Arens regular.

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Jamel Jaber, Marwa Troudi. 2026-10-09. The tensor product of the Fremlin tensor product of Banach lattice algebras. https://arxiv.org/abs/2610.13734

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