Geodesic
Ergodic Dilation of a Quantum Dynamical System
Pandiscia, Carlo1
1 Universitá degli Studi di Roma “Tor Vergata”, Dipartimento di Ingegneria Elettronica, via del Politecnico, 00133 Roma, Italia
Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 77-93.

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Using the Nagy dilation of linear contractions on Hilbert space and the Stinespring’s theorem for completely positive maps, we prove that any quantum dynamical system admits a dilation in the sense of Muhly and Solel which satisfies the same ergodic properties of the original quantum dynamical system.

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DOI : 10.5802/cml.14
Classification : 46L07, 46L55, 46L57
Keywords: Quantum Markov process, completely positive maps, Nagy dilation, ergodic state.

Pandiscia, Carlo 1

1 Universitá degli Studi di Roma “Tor Vergata”, Dipartimento di Ingegneria Elettronica, via del Politecnico, 00133 Roma, Italia 
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Pandiscia, Carlo. Ergodic Dilation of a Quantum Dynamical System. Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 77-93. doi : 10.5802/cml.14. http://geodesic.mathdoc.fr/articles/10.5802/cml.14/

[1] L. Accardi and C. Cecchini. Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Ana., 45:245–273, 1982. | Zbl | DOI | MR

[2] W. Arveson. Non commutative dynamics and Eo-semigroups, Monograph in mathematics, Springer-Verlag, 2003. | DOI | MR

[3] B.V. Bath and K.R. Parthasarathy. Markov dilations of nonconservative dynamical semigroups and quantum boundary theory, Ann. I.H.P. sec. B, 31(4):601–651, 1995. | Zbl | mathdoc-id

[4] D. E. Evans and J. T. Lewis. Dilations of dynamical semi-groups, Comm. Math. Phys., 50(3):219–227, 1976. | Zbl | DOI | MR

[5] A. Frigerio, V.Gorini, A. Kossakowski and M. Verri. Quantum detailed balance and KMS condition, Commun. Math. Phys., 57:97–110, 1977. | Zbl | DOI | MR

[6] B. Kümmerer. Markov dilations on W*-algebras, J. Funct. Ana., 63:139–177, 1985. | Zbl | DOI | MR

[7] W.A. Majewski. On the relationship between the reversibility of dynamics and balance conditions, Ann. I. H. P. sec. A, 39(1):45–54, 1983. | Zbl | mathdoc-id

[8] P.S. Muhly and B. Solel. Quantum Markov Processes (correspondeces and dilations), Int. J. Math., 13(8):863–906, 2002. | Zbl | DOI | MR

[9] B.Sz. Nagy and C. Foiaş. Harmonic analysis of operators on Hilbert space, Regional Conf. Ser. Math., 19, 1971. | Zbl

[10] C. Niculescu, A. Ströh and L.Zsidó. Noncommutative extensions of classical and multiple recurrence theorems, J. Oper. Th., 50:3–52, 2002. | Zbl

[11] V.I. Paulsen. Completely bounded maps and dilations, Pitman Res. Notes Math. 146, Longman Scientific & Technical, 1986. | Zbl | DOI

[12] M. Skeide. Dilation theory and continuous tensor product systems of Hilbert modules, in: PQ-QP: Quantum Probability and White Noise Analysis XV, World Scientific, 2003. | Zbl | DOI

[13] F. Stinesring. Positive functions on C* algebras, Proc. Amer. Math. Soc., 6:211–216, 1955. | DOI | MR

[14] L. Zsido. Personal communication, 2008.

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