BMO-estimates for non-commutative vector valued Lipschitz functions

Abstract

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 $f:\mathbb{R}\to \mathbb{R}$ 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 $f:{\mathbb{R}}^n\to \mathcal{N}$ by imposing Hörmander-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.

To appear in the Journal of Functional Analysis