Completely bounded norms of k$k$‐positive maps.

Given an operator system S$\mathcal {S}$, we define the parameters rk(S)$r_k(\mathcal {S})$ (resp. dk(S)$d_k(\mathcal {S})$) defined as the maximal value of the completely bounded norm of a unital k$k$‐positive map from an arbitrary operator system into S$\mathcal {S}$ (resp. from S$\mathcal {S}$ into an arbitrary operator system). In the case of the matrix algebras Mn$\mathsf {M}_n$, for 1⩽k⩽n$1 \leqslant k \leqslant n$, we compute the exact value rk(Mn)=2n−kk$r_k(\mathsf {M}_n) = \frac{2n-k}{k}$ and show upper and lower bounds on the parameters dk(Mn)$d_k(\mathsf {M}_n)$. Moreover, when S$\mathcal {S}$ is a finite‐dimensional operator system, adapting results of Passer and the fourth author [J. Operator Theory 85 (2021), no. 2, 547–568], we show that the sequence (rk(S))$(r_k(\mathcal {S}))$ tends to 1 if and only if S$\mathcal {S}$ is exact and that the sequence (dk(S))$(d_k(\mathcal {S}))$ tends to 1 if and only if S$\mathcal {S}$ has the lifting property. [ABSTRACT FROM AUTHOR]