Title: $H^\infty$-calculus for semigroup generators on BMO,
Abstract: 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 BMO$(\sqrt L)$ for any angle $\eta>\pi/2$, provided the semigroup satisfies
Bakry-Emry's $\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.