TITLE: Positive but not completely positive?
QUESTION [16 upvotes]: 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?

REPLY [6 votes]: A Jordan homomorphism between C*-algebras is always positive, but it is completely positive only if it is a homomorphism. To get some examples, say $C_r^*(G)$ is the reduced C*-algebra of a discrete group or groupoid $G$. Then the map $g\mapsto g^{-1}$  (a generalized transpose) induces  a Jordan homomorphism on $C_r^*(G)$ which is positive but not completely positive.