  In this Article we analyze the algebraic properties of the asymptotic
dynamics of finite-dimensional open quantum systems in the Heisenberg picture.
In particular, a natural product (Choi-Effros product) can be defined in the
asymptotic regime. Motivated by this structure, we introduce a new space called
the Choi-Effros decoherence-free algebra. Interestingly, this space is both a
C* -algebra with respect to the composition product, and a B* -algebra with
respect to the Choi-Effros product. Moreover, such space admits a direct-sum
decomposition revealing a clear relationship with the attractor subspace of the
dynamics. In particular, the equality between the attractor subspace and the
Choi-Effros decoherence-free algebra is a necessary and sufficient condition
for a faithful dynamics. Finally, we show how all the findings do not rely on
complete positivity but on the much weaker Schwarz property.
