Absolute dilations of ucp self-adjoint Fourier multipliers: the non unimodular case

Let $\varphi$ be a normal semi-finite faithful weight on a von Neumann algebra $A$,let $(\sigma^\varphi_r)_{r\in{\mathbb R}}$ denote the modular automorphism group of $\varphi$, and let $T\colon A\to A$ be a linear map. We say that $T$ admits an absolute dilation if there exist another von Neumann algebra $M$ equipped with a normal semi-finite faithful weight $\psi$, a $w^*$-continuous, unital and weight-preserving $*$-homomorphism $J\colon A\to M$ such that $\sigma^\psi\circ J=J\circ \sigma^\varphi$, as well as a weight-preserving $*$-automorphism $U\colon M\to M$ such that $T^k={\mathbb E}_JU^kJ$ for all integer $k\geq 0$, where ${\mathbb E}_J\colon M\to A$ is the conditional expectation associated with $J$. Given any locally compact group $G$ and any real valued function $u\in C_b(G)$, we prove that if $u$ induces a unital completely positive Fourier multiplier $M_u\colon VN(G) \to VN(G)$, then $M_u$ admits an absolute dilation. Here $VN(G)$ is equiped with its Plangherel weight $\varphi_G$. This result had been settled by the first named author in the case when $G$ is unimodular so the salient point in this paper is that $G$ may be non unimodular, and hence $\varphi_G$ may not be a trace. The absolute dilation of $M_u$ implies that for any $1<p<\infty$, the $L^p$-realization of $M_u$ can be dilated into an isometry acting on a non-commutative $L^p$-space. We further prove that if $u$ is valued in $[0,1]$, then the $L^p$-realization of $M_u$ is a Ritt operator with a bounded $H^\infty$-functional calculus.