  A fundamental aspect of the quantum-to-classical limit is the transition from
a non-commutative algebra of observables to commutative one. However, this
transition is not possible if we only consider unitary evolutions. One way to
describe this transition is to consider the Gamow vectors, which introduce
exponential decays in the evolution. In this paper, we give two mathematical
models in which this transition happens in the infite time limit. In the first
one, we consider operators acting on the space of the Gamow vectors, which
represent quantum resonances. In the second one, we use an algebraic formalism
from scattering theory. We construct a non-commuting algebra which commutes in
the infinite time limit.
