We construct Markov semi-groups $\mathcal{ T}$ and associated BMO-spaces on a finite von Neumann algebra $(\mathcal{ M}, \tau)$ and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that for any $A \in \mathcal{ M}$ self-adjoint and $f: \mathbb{ R} \rightarrow \mathbb{ R}$ Lipschitz there is a Markov semi-group $\mathcal{ T}$ such that for $x \in \mathcal{ M}$, \[ \Vert [f(A), x] \Vert_{ { \rm BMO}(\mathcal{ M}, \mathcal{ T})} \leq c_{ abs} \Vert f' \Vert_\infty \Vert [A, x] \Vert_\infty. \] We obtain an analogue of this result for more general von Neumann valued-functions $f: \mathbb{ R}^n \rightarrow \mathcal{ N}$ by imposing H\"ormander-Mikhlin type assumptions on $f$. In establishing these result we show that Markov dilations of Markov semi-groups have certain automatic continuity properties. We also show that Markov semi-groups of double operator integrals admit (standard and reversed) Markov dilations.
Categories: math.OA math.FA