Let $B\_1$ and $B\_2$ be $C^\*$-algebras. Let $U\_1, \ldots, U\_n$ be some unitaries in $B\_1.$ We consider the operator system $S$ spanned by $U\_iU\_j^\*.$


Let $\phi: S \rightarrow B\_2.$


Given that the block matrix $[\phi(U\_iU\_j^\*)]\_{1 \leq i,j\leq n}$ is positive semidefinite, can we conclude that $\phi$ is completely positive?



Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries
The answer is no :
Let $U\_1 = I$, where $I$ is the identity and $U\_2$ linear independent to $I$ such that $U\_2^\* + U\_2 \ge 0$ .
Then choose $\phi(U\_1) = I$ and $\phi(U\_2) = -I$ .


