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