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Communications in Mathematical Physics

, Volume 185, Issue 1, pp 129–154 | Cite as

q-Gaussian Processes: Non-commutative and Classical Aspects

  • M. Bożejko
  • B. Kümmerer
  • R. Speicher

Abstract:

We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) \(\)– where the a t fulfill the q-commutation relations \(\) for some covariance function \(\)– equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

Keywords

Covariance Hilbert Space Large Class Covariance Function Vacuum Expectation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • M. Bożejko
    • 1
  • B. Kümmerer
    • 2
  • R. Speicher
    • 3
  1. 1.Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland.¶E-mail: bozejko@math.uni.wroc.plPL
  2. 2.Mathematisches Institut A, Pfaffenwaldring 57, D-70569 Stuttgart, Germany.¶E-mail: kuem@mathematik.uni-stuttgart.de DE
  3. 3.Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany.¶E-mail: roland.speicher@urz.uni-heidelberg.deDE

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