Noncommutative Cotlar Identities for Groups Acting on Tree-Like Structures

Let T m Tₘ upper T Subscript m be a noncommutative Fourier multiplier. In recent work, Mei and Ricard introduced a noncommutative analogue of Cotlar's identity to prove that certain multipliers are bounded on the noncommutative L p Lₚ upper L Subscript p -spaces of a free group. Here, we study Cotlar-type identities in full generality, giving a closed characterization for them in terms of m : ( m ( g h ) − m ( g ) ) ( m ( g − 1 ) − m ( h ) ) = 0 , ∀ g ∈ G ∖ { e } , h ∈ G . align*( m(g h) - m(g) ) \, ( m(g⁻¹) - m(h) ) = 0, \; ∀ g ∈ G \, h ∈ G. align* Using a geometric argument, we prove that if X is a tree —or more generally an R R double struck upper R -tree— on which G G normal upper G </jats:inline-fo