I would review results about construction and ergodicity
of stochastic dynamics (of jump type) on inductive limit C^* algebra
which satisfies the detailed balance condition in a Hilbert space
associated to a nontrivial state. Additionally I should say few words
about the set-up for Hypercontractivity and Logarithmic Sobolev inequality
in noncommutative L_p spaces (including corresponding quantum entropy and
- using an idea of particles structure - providing a class examples
states for infinite quantum spin system for which the Logarithmic Sobolev
inequality can be shown).