# Is a linear functional which is positive on linearly generating set of projections positive?

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?

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 $$\varphi(0,1)=-1$$.
• Thank you for your answer! You are absolutely right. As you may have guessed, this isn't quite the question I intended to ask (I am interested in the case where $A$ is concretely represented and $P$ is a complemented lattice). But I am new to stack exchange, so I'm not sure how to proceed. Is the appropriate procedure to accept this answer and ask a new question, or should I edit this one? Thank you again for your help! – User190212 Feb 14 at 0:31
• No problem :)  – Martin Argerami Feb 14 at 0:52
• What if $A$ is simple ? – Epsilon Feb 14 at 8:31