DYNAMICS ON NONCOMMUTATIVE ORLICZ SPACES

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 Det ailed 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].