Dilations of Markovian semigroups of Fourier multipliers on locally compact groups

Arhancet, Cédric

doi:10.1090/proc/14938

Dilations of Markovian semigroups of Fourier multipliers on locally compact groups
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Proc. Amer. Math. Soc. 148 (2020), 2551-2563 Request permission

Abstract:

We prove that any weak* continuous semigroup $(T_t)_{t \geqslant 0}$ of Markov Fourier multipliers acting on a group von Neumann algebra $\mathrm {VN}(G)$ associated to a locally compact group $G$ can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. Our construction relies on probabilistic tools and is even new for the group $\mathbb {R}^n$. Our results imply the boundedness of McIntosh’s $\mathrm {H}^\infty$ functional calculus of the generators of these semigroups on the associated noncommutative $\mathrm {L}^p$-spaces.

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