Free Hilbert transforms

Abstract

We study Fourier multipliers of Hilbert transform type on free groups. We prove that they are completely bounded on noncommutative $L^p$-spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This implies that the decomposition of the free group ${\mathbf{F}}_{\infty }$ into reduced words starting with distinct free generators is completely unconditional in $L^p$. We study the case of Voiculescu’s amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness problem posed by Ozawa, a length-independent estimate for Junge–Parcet–Xu’s free Rosenthal’s inequality, a Littlewood–Paley–Stein-type inequality for geodesic paths of free groups, and a length reduction formula for $L^p$-norms of free group von Neumann algebras.