A note on the order continuity of the norm of \(E\widetilde\otimes_ mF\) (Q1323129)

Nicolae Popa studied permanence properties of tensor products of Banach lattices. By representing a Banach lattice with order continuous norm and a weak order unit as a Banach function space, Popa proved that if \(E\) and \(F\) are Banach lattices with order continuous norm, then the completion \(E\widetilde\otimes_ m F\) of \(E\otimes F\) with respect to the \(m\)-norm has order continuous norm. We use Freudenthal's spectral theorem and de Pagter's component theorem to give a direct proof of Popa's result.