Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• Main Author: Skeide, Michael
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211717
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations. Michael Skeide. SIGMA 18 (2022), 071, 8 pages

We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on 𝓑(𝐺) (𝐺 a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of 𝐸₀-semigroups on 𝓑(𝐻) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general 𝐶*-algebras have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here to make the result available without any appeal to Hilbert modules.)