Speaker：Adrián González Pérez (Autonomous University of Madrid)

Time：2026-9-24 16:00

Location：Conference Room S102 at Experiment Building at Haiyun Campus

Abstract:

In the 80's unpublished work by Haagerup and Gilbert and later of Bozejko and Fendler, showed that, given a function $m: G \to \mathbf{C}$ on a group, the boundedness of its Fourier multiplier on the group von Neumann algebra $L(G)$ and the boundedness of its associated Herz-Schur multiplier acting on the compact operators $K(L_2 G)$ coincide. This result was later extended by Neuwirth/Ricard and Caspers/de la Salle in the case of $L_p$-spaces of $L(G)$ with the extra assumption that $G$ is amenable. A natural question that was left open was whether the equality of norms still holds for nonamenable groups. In particular, it was believed that this was the case when G was a reductive algebraic group and the functions were $K$-biinvariant.

We will present radial counterexamples on the free group and K-biinvariant counterexamples on algebraic groups. The main ingredient are martingale transform techniques and word length decompositions that allows us to transfer the complete boundedness of Fourier multipliers on the $L^p$-space of the torus to the boundedness on Schatten p-classes of radial Schur multiplier on trees. As a consequence, we can construct explicit symbols $m: \mathbf{F}_2 \to \mathbf{C}$ inducing unbounded Fourier multipliers on $L^p(\widehat{\mathbf{F}}_2)$, but whose associated Herz-Schur multipliers are bounded on $S_p$. This result can then be extended, via cocycle induction, to algebraic groups.This is joint work with Javier Parcet, Jorge Pérez García and Simeng Wang. Key insights were provided by Chatgpt-5.6-Sol.