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 $n\to \infty$ ?



Converegence of modulus in nocommutative $L_p$-spaces
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**](http://dx.doi.org/10.7900/jot.2013dec20.2021), J. Oper.
>  Theory 73, No. 2, 361-384 (2015).
>  [ZBL1389.47063](https://zbmath.org/?q=an:1389.47063).
> 
> 
> 


