Free Hilbert transforms (Q2403210)

The Hilbert transform appears as the key tool to define conjugate functions in an abstract setting such as for Dirichlet algebras. In operator algebras, it shows up through Arveson's concept of maximal subdiagonal algebra of a von Neumann algebra. Its \(L^p\)-boundedness is well known and the weak-type \((1,1)\) estimate was obtained by \textit{N. Randrianantoanina} in [J. Aust. Math. Soc., Ser. A 65, No. 3, 388--404 (1998; Zbl 0946.46052)]. In this paper, the authors describe a natural analogue of the Hilbert transform in the context of amalgamated free products of von Neumann algebras. Indeed, their approach is from a different viewpoint than Arveson's and is motivated by a question in the theory of \(L^p\)-Herz-Schur multipliers.