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