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

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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BibTeX RIS

BibTeX

@misc{jaber2026tensor,
  title = {{The tensor product of the Fremlin tensor product of Banach lattice algebras}},
  author = {Jamel Jaber and Marwa Troudi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13734},
  eprint = {2610.13734},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13734},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The tensor product of the Fremlin tensor product of Banach lattice algebras
AU  - Jamel Jaber
AU  - Marwa Troudi
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13734
UR  - https://arxiv.org/abs/2610.13734
AB  - 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.
ER  -

Abstract

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