Dilations of Markovian semigroups of measurable Schur multipliers

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 $\mathrm {B}({\mathrm {L}}^2(X))$ bounded operators Hilbert ${\mathrm {L}}^2(X)$ , where X is a suitable measure space, can be dilated by group Markov $*$ -automorphisms bigger von Neumann algebra. We also construct dilation these semigroups. Our results imply boundedness McIntosh’s {H}}^\infty $ functional calculus generators semigroups associated Schatten spaces and some interpolation connected to {BMO}}$ -spaces. give an answer question Steen, Todorov, Turowska multipliers.