Dilations of markovian semigroups of measurable Schur multipliers

Abstract

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t{)}_{t\ge 0}$ of selfadjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm{B}({\mathrm{L}}^2(X))$ of bounded operators on the Hilbert space ${\mathrm{L}}^2(X)$, where $X$ is a suitable measure space, can be dilated by a weak* continuous group of Markov $\ast$-automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to $\mathrm{BMO}$-spaces. We also give an answer to a question of Steen, Todorov and Turowska on completely positive continuous Schur multipliers.

arXiv comment22 pages, final version