Dynamics on Noncommutative Orlicz Spaces

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 〈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].