2024/06/10 by Duquet, Charles, Merdy, Christian Le
#43A22 #46L51 #47A20 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2406.06074
Let φ be a normal semi-finite faithful weight on a von Neumann algebra A,let (σφr)_r∈\mathbb R denote the modular automorphism group of φ, and let T\colon 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\colon A→ M such that σψ∘ J=J∘ σφ, as well as a weight-preserving *-automorphism U\colon M→ M such that Tk=\mathbb EJUkJ for all integer k≥ 0, where \mathbb EJ\colon 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\colon 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