Let $1\leq p<\infty.$ Let $\mathcal M$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau.$ Let $L_p(\mathcal M,\tau)$ be the associated noncommutative $L_p$space. Let $(x_n)$ be a sequence in $L_p(\mathcal M,\tau)$ such that $\x_nx\_p\to 0$ as $n\to\infty$ for some $x\in L_p(\mathcal M,\tau).$ Is it true that $\x_nx\_p\to 0$ as $n\to \infty$ ?
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This seems false for $p=1$, see the following:
Caspers, M.; Potapov, D.; Sukochev, F.; Zanin, D., Weak type estimates for the absolute value mapping, J. Oper. Theory 73, No. 2, 361384 (2015). ZBL1389.47063.

1$\begingroup$ The results in that paper would solve your question for the other $p$ $\endgroup$ – Adrián GonzálezPérez Aug 20 '19 at 7:44

2$\begingroup$ The modulus map is continuous for $p=1$ as well (although it's not Lipschitz), by Borchers and also by Kosaki [JFA 59 (1984), 123131]. $\endgroup$ – Narutaka OZAWA Aug 28 '19 at 23:22