Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 071, 8 pages      arXiv:0809.3538
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations

Michael Skeide
Università degli Studi del Molise, Dipartimento di Economia, Via de Sanctis, 86100 Campobasso, Italy

Received February 23, 2022, in final form September 23, 2022; Published online October 03, 2022

We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on $\mathscr{B}(G)$ ($G$ 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 $E_0$-semigroups on $\mathscr{B}(H)$ 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 in order to make the result available without any appeal to Hilbert modules.)

Key words: quantum dynamics; quantum probability; quantum Markov semigroups; dilations; product systems.

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  1. Alevras A., The standard form of an $E_0$-semigroup, J. Funct. Anal. 182 (2001), 227-242.
  2. Arveson W., Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (1989), iv+66 pages.
  3. Arveson W., The index of a quantum dynamical semigroup, J. Funct. Anal. 146 (1997), 557-588, arXiv:funct-an/9705005.
  4. Arveson W., Noncommutative dynamics and $E$-semigroups, Springer Monogr. Math., Springer-Verlag, New York, 2003.
  5. Arveson W., Kishimoto A., A note on extensions of semigroups of $*$-endomorphisms, Proc. Amer. Math. Soc. 116 (1992), 769-774.
  6. Bhat B.V.R., An index theory for quantum dynamical semigroups, Trans. Amer. Math. Soc. 348 (1996), 561-583.
  7. Bhat B.V.R., Cocycles of CCR flows, Mem. Amer. Math. Soc. 149 (2001), x+114 pages.
  8. Bhat B.V.R., Liebscher V., Skeide M., Subsystems of Fock need not be Fock: spatial CP-semigroups, Proc. Amer. Math. Soc. 138 (2010), 2443-2456, arXiv:0804.2169.
  9. Bhat B.V.R., Skeide M., Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 519-575.
  10. Chebotarev A.M., Fagnola F., Sufficient conditions for conservativity of minimal quantum dynamical semigroups, J. Funct. Anal. 153 (1998), 382-404, arXiv:funct-an/9711006.
  11. Fowler N.J., Free $E_0$-semigroups, Canad. J. Math. 47 (1995), 744-785.
  12. Goswami D., Sinha K.B., Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra, Comm. Math. Phys. 205 (1999), 377-403.
  13. Hellmich J., Köstler C., Kümmerer B., Noncommutative continuous Bernoulli sifts, arXiv:math.OA/0411565.
  14. Hudson R.L., Parthasarathy K.R., Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), 301-323.
  15. Hudson R.L., Parthasarathy K.R., Stochastic dilations of uniformly continuous completely positive semigroups, Acta Appl. Math. 2 (1984), 353-378.
  16. Köstler C., Quanten-Markoff-Prozesse und Quanten-Brownsche Bewegungen, Ph.D. Thesis, University of Stuttgart, 2000.
  17. Kümmerer B., Speicher R., Stochastic integration on the Cuntz algebra $O_\infty$, J. Funct. Anal. 103 (1992), 372-408.
  18. Lindsay J.M., Quantum stochastic analysis - an introduction, in Quantum Independent Increment Processes. I, Lecture Notes in Math., Vol. 1865, Springer, Berlin, 2005, 181-271.
  19. Parthasarathy K.R., An introduction to quantum stochastic calculus, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1992.
  20. Powers R.T., A nonspatial continuous semigroup of $*$-endomorphisms of ${\mathfrak B}({\mathfrak H})$, Publ. Res. Inst. Math. Sci. 23 (1987), 1053-1069.
  21. Powers R.T., New examples of continuous spatial semigroups of $\ast$-endomorphisms of $\mathfrak B(\mathfrak H)$, Internat. J. Math. 10 (1999), 215-288.
  22. Shalit O., Skeide M., CP-semigroups and dilations, subproduct systems and superproduct systems: The multi-parameter case and beyond, Dissertationes Math., to appear, arXiv:2003.05166.
  23. Skeide M., Quantum stochastic calculus on full Fock modules, J. Funct. Anal. 173 (2000), 401-452.
  24. Skeide M., Dilations, product systems and weak dilations, Math. Notes 71 (2002), 836-843.
  25. Skeide M., The index of (white) noises and their product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 617-655, arXiv:math.OA/0601228.
  26. Skeide M., A simple proof of the fundamental theorem about Arveson systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 305-314, arXiv:math.OA/0602014.
  27. Skeide M., Spatial $E_0$-semigroups are restrictions of inner automorphism groups, in Quantum Probability and Infinite Dimensional Analysis, QP-PQ: Quantum Probab. White Noise Anal., Vol. 20, World Sci. Publ., Hackensack, NJ, 2007, 348-355, arXiv:math.OA/0509323.
  28. Skeide M., Constructing proper Markov semigroups for Arveson systems, J. Math. Anal. Appl. 386 (2012), 796-800, arXiv:1006.2211.
  29. Skeide M., Classification of $E_0$-semigroups by product systems, Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798.
  30. Skeide M., Free product systems generated by spatial tensor product systems, in preparation.
  31. Tsirelson B., Unitary Brownian motions are linearizable, arXiv:math.PR/9806112.
  32. Tsirelson B., Scaling limit, noise, stability, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., Vol. 1840, Springer, Berlin, 2004, 1-106, arXiv:math.PR/0301237.
  33. Tsirelson B., On automorphisms of type II Arveson systems (probabilistic approach), New York J. Math. 14 (2008), 539-576, arXiv:math.OA/0411062.

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