The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator

Let T be a quantum Markov semigroup on B(h) with a faithful normal invariant state ρ. The decoherence-free subalgebra N(T) of is the biggest subalgebra of B(h) where the completely positive maps T_t act as homomorphisms. When T is the minimal semigroup whose generator is represented in a generalised GKSL form, with possibly unbounded operators H, Lℓ, we show that coincides with a generalised commutator under some natural regularity conditions. We give examples of quantum Markov semigroups , with h infinite-dimensional, having a non-trivial decoherence-free subalgebra. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ.