A characterization of absolutely dilatable Schur multipliers

Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T\colon M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M'$ equipped with a nsf trace, a $w^*$-continuous trace preserving unital $*$-homomorphim $J\colon M\to M'$ and a trace preserving $*$-automomorphim $U\colon M'\to M'$ such that $T^k=E U^k J$ for all integer $k\geq 0$, where $E\colon M'\to M$ is the conditional expectation associated with $J$. Given a $\sigma$-finite measure space $(\Omega,\mu)$, we characterize bounded Schur multipliers $\phi\in L^\infty(\Omega^2)$ such that the Schur multiplication operator $T_\phi\colon B(L^2(\Omega))\to B(L^2(\Omega))$ is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra $N$ with a separable predual, equipped with a normalized normal faithful trace $\tau_N$, and of a $w^*$-continuous essentially bounded function $d\colon\Omega\to N$ such that $\phi(s,t)=\tau_N(d(s)^*d(t))$ for almost every $(s,t)\in\Omega^2$.