$H^infty$-calculus for semigroup generators on BMO

We prove that the negative infinitesimal generator $L$ of a semigroup of positive contractions on $L^infty$ has a bounded $H^infty(S_eta^0)$-calculus on the associated Poisson semigroup-BMO space for any angle $eta>pi/2$, provided the semigroup satisfies Bakry-Emrys $Gamma_2 $ criterion. Our arguments only rely on the properties of the underlying semigroup and works well in the noncommutative setting. A key ingredient of our argument is a quasi monotone property for the subordinated semigroup $T_{t,alpha}=e^{-tL^alpha},0<alpha<1$, that is proved in the first half of the article.