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.