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

Let φ be a normal semi-finite faithful weight on a von Neumann algebra A,let (σ^φᵣ)_r∈ R denote the modular automorphism group of φ, and let T A→ 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 ψ, a w^*-continuous, unital and weight-preserving $*$-homomorphism J A→ M such that σ^ψ∘ J=J∘ σ^φ, as well as a weight-preserving $*$-automorphism U M→ M such that Tᵏ= EJUᵏJ for all integer k≥ 0, where EJ M→ A is the conditional expectation associated with J. Given any locally compact group G and any real valued function u∈ Cb(G), we prove that if u induces a unital completely positive Fourier multiplier Mᵤ VN(G) → VN(G), then Mᵤ admits an absolute dilation. Here $VN(G)$ is equiped with its Plangherel weight φG. 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 φG may not be a trace. The absolute dilation of Mᵤ implies that for any 1<p<∞, the Lᵖ-realization of Mᵤ can be dilated into an isometry acting on a non-commutative Lᵖ-space. We further prove that if u is valued in $[0,1]$, then the Lᵖ-realization of Mᵤ is a Ritt operator with a bounded H^∞-functional calculus.