The only example I know of a positive map which is not completely positive is the transpose map on $M\_n(\mathbb{C})$. Of course, one can come up with minor perturbations of this (compose it with, or add it to, a completely positive map, etc). Are there other known examples?



What are known examples of positive but not completely positive maps?
In Theorem 4.6 of their paper


<http://www.univie.ac.at/nuhag-php/bibtex/open_files/deha85_CanniereHaagerup.pdf>


de Canni`ere and Haagerup construct an explicit sequence of finitely supported functions on the free group $\mathbb{F}\_N$ ($N\geq 2$), defining positive multipliers of the reduced C\*-algebra $C^\*\_r(\mathbb{F}\_N)$, and such that the corresponding sequence of multipliers converges strongly to the identity of $C^\*\_r(\mathbb{F}\_N)$. As $C^\*\_r(\mathbb{F}\_N)$ is non-nuclear, it does not have the completely positive approximation property, so only finitely many of these multipliers are completely positive.


