Cotlar identities for groups acting on tree like structures

The Hilbert transform H is a basic example of a Fourier multiplier, and Riesz proved that H is a bounded operator on Lp(T) for all p between 1 and infinity.  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.  This is joint work with Adrián González and Javier Parcet.