# Converegence of modulus in nocommutative $L_p$-spaces

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 $$\|x_n-x\|_p\to 0$$ as $$n\to\infty$$ for some $$x\in L_p(\mathcal M,\tau).$$ Is it true that $$\||x_n|-|x|\|_p\to 0$$ as $$n\to \infty$$ ?

This seems false for $$p=1$$, see the following:
• The results in that paper would solve your question for the other $p$ – Adrián González-Pérez Aug 20 '19 at 7:44
• The modulus map is continuous for $p=1$ as well (although it's not Lipschitz), by Borchers and also by Kosaki [JFA 59 (1984), 123-131]. – Narutaka OZAWA Aug 28 '19 at 23:22