Let [Formula: see text] be a uniformly continuous quantum Markov semigroup on [Formula: see text] with generator represented in a standard GKSL form [Formula: see text] and a faithful normal invariant state ρ. In this note we give new algebraic conditions for proving that [Formula: see text] converges towards a steady state, possibly different from ρ. Indeed, we show that this happens whenever the commutator of [Formula: see text] (i.e. its fixed point algebra) coincides with the commutator of [Formula: see text] (where δH(X) = [H, X]) for some n ≥ 1. As an application we discuss the convergence to the unique invariant state of a spin chain model.