Abstract
We exhibit a Hamel basis for the concrete \(*\)-algebra \({\mathfrak {M}}_o\) associated to monotone commutation relations realised on the monotone Fock space, mainly composed by Wick ordered words of annihilators and creators. We apply such a result to investigate spreadability and exchangeability of the stochastic processes arising from such commutation relations. In particular, we show that spreadability comes from a monoidal action implementing a dissipative dynamics on the norm closure \(C^*\)-algebra \({\mathfrak {M}}=\overline{{\mathfrak {M}}_o}\). Finally, we determine the structure of the set of exchangeable and spreadable monotone stochastic processes, by showing that both coincide with the stationary ones.
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Acknowledgements
The authors acknowledge the support of the Italian INDAM-GNAMPA. The first and the second named authors kindly acknowledge also the Department of Physics of the University of Pretoria, where part of the results of the present paper was obtained, and in particular R. Duvenhage for the hospitality and financial support. Finally, the authors are grateful to an anonymous referee whose comments improved the presentation of the paper.
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Communicated by Claude Alain Pillet.
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Crismale, V., Fidaleo, F. & Griseta, M.E. Wick Order, Spreadability and Exchangeability for Monotone Commutation Relations. Ann. Henri Poincaré 19, 3179–3196 (2018). https://doi.org/10.1007/s00023-018-0706-2
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DOI: https://doi.org/10.1007/s00023-018-0706-2