The Hilbert transform $H$ is a basic example of Fourier multipliers.  Its behaviour on Fourier series is the following:

$$
\sum_{n\in \mathbb{Z}}a_n e^{inx} \longmapsto \sum_{n\in \mathbb{Z}}m(n)a_n e^{inx},
$$
with $m(n)=-i\,{\rm sgn} (n)$.
Riesz proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$.
We study  Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative $L_p$ spaces.
The pioneering work in this direction is due to Mei and Ricard who proved $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on tree-like structures. This class of groups includes amalgamated free products, HNN extensions, left orderable groups and many others.

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Joint work with Adri\'an Gonz\'alez and Javier Parcet.