Completely bounded norms of kk‐positive maps

Abstract Given an operator system , we define the parameters (resp. ) defined as the maximal value of the completely bounded norm of a unital ‐positive map from an arbitrary operator system into (resp. from into an arbitrary operator system). In the case of the matrix algebras , for , we compute the exact value and show upper and lower bounds on the parameters . Moreover, when 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 tends to 1 if and only if is exact and that the sequence tends to 1 if and only if has the lifting property.