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Dilations of Markovian semigroups of measurable Schur multipliers
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 05 April 2023, pp. 774-797
- Print publication:
- June 2024
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- Article
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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.
