Absolute dilation of Fourier multipliers

Let ${\mathcal M}$ be a von Neumann algebra equipped with a normal semifinite faithful (nsf) trace. We say that an operator $T :{\mathcal M}\to {\mathcal M}$ is absolutely dilatable if there exist another von Neumann algebra $M$ with an nsf trace, a unital normal trace preserving $\ast$-homomorphism $J: {\mathcal M} \to M$, and a trace preserving $\ast$-automorphism $U: M \to M$ such that $T^k = {\mathbb E}_J U^k J \quad \text{for all } k \geq 0,$ where ${\mathbb E}_J: M \to {\mathcal M}$ is the conditional expectation associated with $J$. For a discrete amenable group $G$ and a function $u:G\to\mathbb{C}$ inducing a unital completely positive Fourier multiplier $M_u: VN(G) \to VN(G)$, we establish the following transference theorem: the operator $M_u$ admits an absolute dilation if and only if its associated Herz-Schur multiplier does. From this result, we deduce a characterization of Fourier multipliers with an absolute dilation in this setting. Building on the transference result, we construct the first known example of a unital completely positive Fourier multiplier that does not admit an absolute dilation. This example arises in the symmetric group ${\mathcal S}_3$, the smallest group where such a phenomenon occurs. Moreover, we show that for every abelian group $G$, every Fourier multiplier always admits an absolute dilation.