Let $A$ be a finite dimensional $C^*$ algebra and $S(A)$ the state space. Let $K\subset A$ be an intersection of $S(A)$ with a vector subspace $J\subset A$ and let $f$ be a positive affine functional on $K$.
I am dealing with the following questions: Can $f$ be extended to a positive linear functional on $A$? And if $F$ is a completely positive map on $J$, under which conditions can it be extended to a completely positive map on $A$? Is there anything else than Arveson's extension theorem?
Is there a good reference where these questions are treated in the finite dimensional case?