Let $A$ be a unital C$^*$ algebra, and suppose there is a set of projections $P \subset \mathcal{P}(A)$ whose linear span is dense in $A$. If $\varphi \in A^*$ has $\varphi(p) \ge 0$ for all $p \in P$, does it follow that $\varphi \ge 0$?
Note that this does hold if every element of $A$ can be approximated in norm by a linear combination of mutually orthogonal projections in $P$ (given any $x^*x \in A_+$, such an approximation for $x$ will lead to an approximation of $x^*x$ by a linear combination with positive coefficients), but is there any reason to believe it in general?
No.
Take $A=\mathbb C^2$, $\varphi(a,b)=2a-b$, and $$ \mathcal P=\{(1,0),(1,1)\}. $$ Then $\operatorname{span}\mathcal P=A$, and $\varphi(1,0)=2$, $\varphi(1,1)=1$. But $\varphi$ is not positive, since $\varphi(0,1)=-1$.