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 〈Lcosh−1, L log(L + 1)〉, 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 (L∞, L1). As a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26].

中文翻译：

---

非对易 Orlicz 空间上的动力学

量子动力学映射是为基于 Orlicz 空间的量子统计物理定义和研究的。这补充了早期的工作 [26]，其中我们为常规系统的统计物理应该适当地基于 Orlicz 空间对 〈Lcosh−1, L log(L + 1)〉 的断言提供了强有力的理由，因为该框架给出更好地描述常规可观察量，并且还允许定义明确的熵函数。在本文中，我们通过解决这样一个系统的动力学问题来“完成”这幅图，正如对应于某种狄利克雷形式的马尔可夫半群所描述的那样（见 [4, 13, 14]）。具体来说，我们表明，即使在最一般的非交换上下文中，满足自然详细平衡条件的完全正马尔可夫映射也规范地承认对一大类量子 Orlicz 空间的作用。这是通过开发新的插值策略来实现的，该策略将此类映射的动作扩展到对 (L∞, L1) 的适当中间空间。因此，我们获得完全正的量子马尔可夫动力学自然扩展到 [26] 中提出的上下文。