Given an operator system $S$, we define the parameters $r_k(S)$ and $d_k(S)$ defined as the maximal value of the completely bounded norm of a unital $k$-positive map from an arbitrary operator system into $S$ and from $S$ into an arbitrary operator system. In the case of the matrix algebras $M_n$, for $1 \le k \le n$, we compute the exact value $r_k(M_n) = \frac{2n-k}{k}$ and obtain upper and lower bounds on the parameters $d_k(M_n)$. When $S$ is a finite- dimensional operator system, adapting results of Passer and Paulsen, we show that the sequence $(r_k(S))$ tends to 1 if and only if $S$ is exact and that the sequence $(d_k(S))$ tends to 1 if and only if $S$ has the lifting property.

This is joint work with G. Aubrun, A. Müller-Hermes, V. Paulsen and M. Rahaman.