Dilations of Markovian semigroups of measurable Schur multipliers

Abstract

Abstract Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geqslant 0}$ of self-adjoint 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 $*$ -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.