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Dilations of Markovian semigroups of measurable Schur multipliers
Part of:
Selfadjoint operator algebras
Methods of category theory in functional analysis
Groups and semigroups of linear operators, their generalizations and applications
General theory of linear operators
Published online by Cambridge University Press: 05 April 2023
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.
Keywords
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
Footnotes
The author acknowledges support by the grant ANR-18-CE40-0021 (project HASCON) of the French National Research Agency ANR.
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