2025/02/25 by Merdy, Christian Le, Zadeh, Safoura
#43A22 #46L51 #47A20 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2502.18011
Let \mathcal M be a von Neumann algebra equipped with a normal semifinite faithful (nsf) trace. We say that an operator T :\mathcal M→ \mathcal M is absolutely dilatable if there exist another von Neumann algebra M with an nsf trace, a unital normal trace preserving ∗-homomorphism J: \mathcal M → M, and a trace preserving ∗-automorphism U: M → M such that Tk = \mathbb EJ Uk J for all k ≥ 0, where \mathbb EJ: M → \mathcal M is the conditional expectation associated with J. For a discrete amenable group G and a function u:G→ℂ inducing a unital completely positive Fourier multiplier Mu: VN(G) → VN(G), we establish the following transference theorem: the operator Mu 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 S3, 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.