# Ergodic Dilation of a Quantum Dynamical System

Carlo Pandiscia^{[1]}

- [1] Universitá degli Studi di Roma “Tor Vergata”, Dipartimento di Ingegneria Elettronica, via del Politecnico, 00133 Roma, Italia

Confluentes Mathematici (2014)

- Volume: 6, Issue: 1, page 77-91
- ISSN: 1793-7434

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topPandiscia, Carlo. "Ergodic Dilation of a Quantum Dynamical System." Confluentes Mathematici 6.1 (2014): 77-91. <http://eudml.org/doc/275495>.

@article{Pandiscia2014,

abstract = {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.},

affiliation = {Universitá degli Studi di Roma “Tor Vergata”, Dipartimento di Ingegneria Elettronica, via del Politecnico, 00133 Roma, Italia},

author = {Pandiscia, Carlo},

journal = {Confluentes Mathematici},

keywords = {Quantum Markov process; completely positive maps; Nagy dilation; ergodic state; quantum Markov process},

language = {eng},

number = {1},

pages = {77-91},

publisher = {Institut Camille Jordan},

title = {Ergodic Dilation of a Quantum Dynamical System},

url = {http://eudml.org/doc/275495},

volume = {6},

year = {2014},

}

TY - JOUR

AU - Pandiscia, Carlo

TI - Ergodic Dilation of a Quantum Dynamical System

JO - Confluentes Mathematici

PY - 2014

PB - Institut Camille Jordan

VL - 6

IS - 1

SP - 77

EP - 91

AB - 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.

LA - eng

KW - Quantum Markov process; completely positive maps; Nagy dilation; ergodic state; quantum Markov process

UR - http://eudml.org/doc/275495

ER -

## References

top- L. Accardi and C. Cecchini. Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Ana., 45:245–273, 1982. Zbl0483.46043MR647075
- W. Arveson. Non commutative dynamics and Eo-semigroups, Monograph in mathematics, Springer-Verlag, 2003. Zbl1032.46001
- 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. Zbl0832.46060MR1355610
- D. E. Evans and J. T. Lewis. Dilations of dynamical semi-groups, Comm. Math. Phys., 50(3):219–227, 1976. Zbl0402.46039MR468878
- A. Frigerio, V.Gorini, A. Kossakowski and M. Verri. Quantum detailed balance and KMS condition, Commun. Math. Phys., 57:97–110, 1977. Zbl0374.46060MR468989
- B. Kümmerer. Markov dilations on W*-algebras, J. Funct. Ana., 63:139–177, 1985. Zbl0601.46062MR803091
- 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. Zbl0519.46068MR715131
- P.S. Muhly and B. Solel. Quantum Markov Processes (correspondeces and dilations), Int. J. Math., 13(8):863–906, 2002. Zbl1057.46050MR1928802
- B.Sz. Nagy and C. Foiaş. Harmonic analysis of operators on Hilbert space, Regional Conf. Ser. Math., 19, 1971. Zbl0201.45003MR275190
- C. Niculescu, A. Ströh and L.Zsidó. Noncommutative extensions of classical and multiple recurrence theorems, J. Oper. Th., 50:3–52, 2002. Zbl1036.46053MR2015017
- V.I. Paulsen. Completely bounded maps and dilations, Pitman Res. Notes Math. 146, Longman Scientific & Technical, 1986. Zbl0614.47006MR868472
- 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. Zbl1050.81043MR2010609
- F. Stinesring. Positive functions on C* algebras, Proc. Amer. Math. Soc., 6:211–216, 1955. Zbl0064.36703MR69403
- L. Zsido. Personal communication, 2008.

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