Talagrand type for noncommutative $L_1$ spaces

Let $(\mathcal{M},\tau )$ be a semi-finite von Neumann algebra. We prove that $L_1(\mathcal{M})$ has Talagrand type $(1,{\psi }_{1,1})$, where ${\psi }_{1,1}(t)=t\mathrm{/}\log (e+t)$. As an consequence, the Schatten trace class $S_1$ has Talagrand type $(1,{\psi }_{1,1})$, thus settled an open problem of Question 2 in Cordero-Erausquin and Eskenazis (2023). Our proof is based on the noncommutative Mazur maps initiated by Ricard.