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We construct Markov semi-groups $\mathcal{T}$ and associated BMO-spaces on afinite von Neumann algebra $(\mathcal{M}, \tau)$ and obtain results forperturbations of commutators and non-commutative Lipschitz estimates. Inparticular, 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 $x \in \mathcal{M}$, \[ \Vert [f(A), x] \Vert_{{\rmBMO}(\mathcal{M}, \mathcal{T})} \leq c_{abs} \Vert f' \Vert_\infty \Vert [A, x]\Vert_\infty. \] We obtain an analogue of this result for more general vonNeumann valued-functions $f: \mathbb{R}^n \rightarrow \mathcal{N}$ by imposingH\"ormander-Mikhlin type assumptions on $f$. In establishing these result weshow that Markov dilations of Markov semi-groups have certain automaticcontinuity properties. We also show that Markov semi-groups of double operatorintegrals admit (standard and reversed) Markov dilations.

BMO-estimates for non-commutative vector valued Lipschitz functions