Completely bounded norms of k $k$ -positive maps

Abstract

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