数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1249-1270.doi: 10.1007/s10473-020-0507-9

• 论文 • 上一篇    下一篇

DYNAMICS ON NONCOMMUTATIVE ORLICZ SPACES

L. E. LABUSCHAGNE1, W. A. MAJEWSKI2   

  1. 1. DSI-NRF CoE in Mathematics and Statistics Science, Focus Area for PAA, Internal Box 209, School of Mathematics and Statistics Science NWU, PVT. BAG X6001, 2520 Potchefstroom, South Africa;
    2. Focus Area for PAA, North-West-University, Potchefstroom, South Africa
  • 收稿日期:2019-07-16 修回日期:2020-01-14 出版日期:2020-10-25 发布日期:2020-11-04
  • 通讯作者: L.E. LABUSCHAGNE E-mail:Louis.Labuschagne@nwu.ac.za
  • 作者简介:W.A. MAJEWSKI,E-mail:fizwam@univ.gda.pl
  • 基金资助:
    The contribution of L. E. Labuschagne is based on research partially supported by the National Research Foundation (IPRR Grant 96128). Any opinion, findings and conclusions or recommendations expressed in this material, are those of the author, and therefore the NRF do not accept any liability in regard thereto.

DYNAMICS ON NONCOMMUTATIVE ORLICZ SPACES

L. E. LABUSCHAGNE1, W. A. MAJEWSKI2   

  1. 1. DSI-NRF CoE in Mathematics and Statistics Science, Focus Area for PAA, Internal Box 209, School of Mathematics and Statistics Science NWU, PVT. BAG X6001, 2520 Potchefstroom, South Africa;
    2. Focus Area for PAA, North-West-University, Potchefstroom, South Africa
  • Received:2019-07-16 Revised:2020-01-14 Online:2020-10-25 Published:2020-11-04
  • Contact: L.E. LABUSCHAGNE E-mail:Louis.Labuschagne@nwu.ac.za
  • Supported by:
    The contribution of L. E. Labuschagne is based on research partially supported by the National Research Foundation (IPRR Grant 96128). Any opinion, findings and conclusions or recommendations expressed in this material, are those of the author, and therefore the NRF do not accept any liability in regard thereto.

摘要: Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces $\langle L^{\cosh - 1}, L\log(L+1)\rangle$, since this framework gives a better description of regular observables, and also allows for a well-defined entropy function. In the present paper we "complete" the picture by addressing the issue of the dynamics of such a system, as described by a Markov semigroup corresponding to some Dirichlet form (see [4, 13, 14]). Specifically, we show that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation strategy for extending the action of such maps to the appropriate intermediate spaces of the pair $\langle L^\infty,L^1\rangle$. As a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26].

关键词: detailed balance, Orlicz space, Markov semigroup, completely positive

Abstract: Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces $\langle L^{\cosh - 1}, L\log(L+1)\rangle$, since this framework gives a better description of regular observables, and also allows for a well-defined entropy function. In the present paper we "complete" the picture by addressing the issue of the dynamics of such a system, as described by a Markov semigroup corresponding to some Dirichlet form (see [4, 13, 14]). Specifically, we show that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation strategy for extending the action of such maps to the appropriate intermediate spaces of the pair $\langle L^\infty,L^1\rangle$. As a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26].

Key words: detailed balance, Orlicz space, Markov semigroup, completely positive

中图分类号: 

  • 46L55