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Dilations of semigroups on von Neumann algebras and noncommutative \(L^{p}\)-spaces.(English)Zbl 07026816

Summary: We prove that any weak* continuous semigroup \((T_t)_{t \geqslant 0}\) of factorizable Markov maps acting on a von Neumann algebra \(M\) equipped with a normal faithful state can be dilated by a group of Markov \(\ast\)-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative \(\mathrm{L}^p\)-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh’s \(\mathrm{H}^\infty\) functional calculus of the generators of these semigroups on the associated noncommutative \(\mathrm{L}^p\)-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of \(\mathbb{R}^n\).

MSC:

47A20 Dilations, extensions, compressions of linear operators
47D03 Groups and semigroups of linear operators
46L51 Noncommutative measure and integration
47D07 Markov semigroups and applications to diffusion processes

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References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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