  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.
