Abstract: The talk is based on a joint work in progress with Magdalena Musat. We study two dilation properties for completely positive unital trace preserving maps (for short, cp.u.t. maps) on (M, tr), where M is a von Neumann algebra and tr is a normal faithful trace state on M. The first property is Kümmerer's Markov dilation property from the 80's, which is equivalent to Anantharaman-Delaroche's factorization property from 2004. For this, we provide, for instance, an example of one-parameter semigroup (T_t)_t >= 0 of cp.u.t. maps on the 4 x 4 matrices such that T_t fails to have the Markov dilation property for all small values of t > 0. The second property is the non-commutative Rota dilation property introduced by Junge, Le Merdy and Xu in 2006. We show that the most natural generalization of Rota's classical dilation theorem to the non-commutative setting does not hold by providing an example of a selfadjoint cp.u.t. map T on the n x n matrices for some large n, such that T2 does not have the non-commutative Rota dilation property.