Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T11:04:32.659Z Has data issue: false hasContentIssue false
  • English
  • Français

Completely positive quantum stochastic convolution cocycles and their dilations

Published online by Cambridge University Press:  01 July 2007

ADAM G. SKALSKI
ADAM G. SKALSKI*
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF. e-mail: a.skalski@lancaster.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C*-hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C*-bialgebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 201 - 219
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Accardi, L.. On the quantum Feynman–Kac formula. Rend. Sem. Mat. Fis. Milano 48 (1978), 135–180.CrossRefGoogle Scholar
[2]Belavkin, V. P.. Quantum stochastic positive evolutions: characterization, construction, dilation. Comm. Math. Phys. 184 (1997), no. 3, 533–566.CrossRefGoogle Scholar
[3]Bloom, W. R. and Heyer, H.. Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter Studies in Mathematics, 20 (Walter de Gruyter & Co., 1995).CrossRefGoogle Scholar
[4]Effros, E. G. and Ruan, Z. J.. Operator Spaces (Oxford University Press, 2000).Google Scholar
[5] YChapovsky, u. and Vainerman, L.. Compact quantum hypergroups. J. Operator Theory 41 (1999), no. 2, 261–289.Google Scholar
[6]Franz, U.. Lévy processes on quantum groups and dual groups. In Quantum Independent Increment Processes, Vol. II: Structure of Quantum Lévy Processes, Classical Probability and Physics, eds. Franz, U. and Schürmann, M.. Lecture Notes in Math. 1866 (Springer-Verlag, 2005).Google Scholar
[7]Franz, U. and M. Schürmann. Lévy processes on quantum hypergroups. In Infinite Dimensional Harmonic Analysis, eds. Heyer, H., Hirai, T. and Obata, N. (Gräbner, 2000), pp. 93–114.Google Scholar
[8]Goswami, D., Lindsay, J. M. and Wills, S. J.. A stochastic Stinespring theorem, Math. Ann. 319 (2001), no. 4, 647–673.CrossRefGoogle Scholar
[9]Goswami, D., Lindsay, J. M., Sinha, K. B. and Wills, S. J.. Dilation of Markovian cocycles on a von Neumann algebra. Pacific J. Math. 211 (2003), 221–247.CrossRefGoogle Scholar
[10]Kalyuzhnyi, A. A.. Conditional expectations on compact quantum groups and new examples of quantum hypergroups. Methods Funct. Anal. Topology 7 (2001), no. 4, 49–68.Google Scholar
[11]Kalyuzhnyi, A. A. and Chapovsky, Yu. A.. A factorization of conditional expectations on Kac algebras and quantum double coset hypergroups. Ukraïn. Mat. Zh. 55 (2003), no. 12, 1669–1677; translation in Ukrainian Math. J. 55 (2003), no. 12, 1994–2005.Google Scholar
[12]Lindsay, J. M.. Quantum stochastic analysis – an introduction. In Quantum Independent Increment Processes, Vol. I: From Classical Probability to Quantum Stochastics, eds. Franz, U. and Schürmann, M.. Lecture Notes in Math. 1865 (Springer-Verlag, 2005).Google Scholar
[13]Lindsay, J. M. and Parthasarathy, K. R.. On the generators of quantum stochastic flows. J. Funct. Anal. 158 (1998), 521–549.CrossRefGoogle Scholar
[14]Lindsay, J. M. and Skalski, A. G.. Quantum stochastic convolution cocycles I. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3 (En hommage à Paul–André Meyer), 581–604.CrossRefGoogle Scholar
[15]Lindsay, J. M. and Skalski, A. G.. Quantum stochastic convolution cocycles—algebraic and C*-algebraic. Banach Center Publ. 73 (2006), 313–324.CrossRefGoogle Scholar
[16]Lindsay, J. M. and Skalski, A. G.. On quantum stochastic differential equations. J. Math. Anal. Appl. 330 (2007), no. 2, 1093–1114.CrossRefGoogle Scholar
[17]Lindsay, J. M. and Skalski, A. G.. Quantum stochastic convolution cocycles II, preprint.Google Scholar
[18]Lindsay, J. M. and Wills, S. J.. Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise. Probab. Theory Related Fields 116 (2000), 505–543.CrossRefGoogle Scholar
[19]Lindsay, J. M. and Wills, S. J.. Markovian cocycles on operator algebras, adapted to a Fock filtration. J. Funct. Anal. 178 (2000), no. 2, 269–305.CrossRefGoogle Scholar
[20]Lindsay, J. M. and Wills, S. J.. Existence of Feller cocycles on a C*-algebra. Bull. London Math. Soc. 33 (2001), no. 5, 613–621.CrossRefGoogle Scholar
[21]Lindsay, J. M. and Wills, S. J.. Homomorphic Feller cocycles on a C*-algebra. J. London Math. Soc. (2) 68 (2003), no. 1, 255–272.CrossRefGoogle Scholar
[22] M. Schürmann. White Noise on Bialgebras. Lecture Notes in Math. 1544 (Springer, 1993).CrossRefGoogle Scholar