  We study factorization and dilation properties of Markov maps between von
Neumann algebras equipped with normal faithful states, i.e., completely
positive unital maps which preserve the given states and also intertwine their
automorphism groups. The starting point for our investigation has been the
question of existence of non-factorizable Markov maps, as formulated by C.
Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov
maps on M_n(C) for all n\geq 3, as well as an example of a one-parameter
semigroup (T(t))_{t\geq 0} of Markov maps on M_4(C) such that T(t) fails to be
factorizable for all small values of t > 0. As applications, we solve in the
negative an open problem in quantum information theory concerning an asymptotic
version of the quantum Birkhoff conjecture, as well as we sharpen the existing
lower bound estimate for the best constant in the noncommutative little
Grothendieck inequality.
