OK, but I don't see that if $x,y\in c(V)$ with $x-y\le 1$, then $f(x-y)\le \lambda$

Yes, I have this book, but I only found extension theorem for an operator system, which is a self adjoint subspace, containing the unit. But I wonder if there is an extension theorem for other situations, e.g. when $J$ is self-adjoint, but does not contain the unit (but maybe a positive invertible element).

I don't think one can find a positive extension for any convex subset. Let $V$ be a convex subset generated by two positive trace 1 elements $\rho_1$ and $\rho_2$. Then for any $f_1,f_2\ge 0$ there is a positive affine functionals on $V$ with $f_1=f(\rho_1)$ and $f_2=f(\rho_2)$. Suppose that $\rho_1\le M\rho_2$ for some $M>1$. Then $M\rho_2-\rho_1$ is positive, but one can always find $f_1,f_2\ge 0$ such that $Mf_2-f_1<0$.

Does it mean that one can do this for any convex subset of $S(A)$?

Yes, thanks very much. For the last 2 questions: I was asking about a cp map $J\to B(H)$, for a Hilbert space $H$, not a functional.

Since $A$ is finite dimensional, the states can be identified with density operator with respect to a fixed trace, that is, positive operators with trace 1. Then $S(A)\subset A$. In the general case, the question has to be formulated in the dual space, but in finite dimensions, this is all the same.