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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?

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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 $\varphi(0,1)=-1$.

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  • $\begingroup$ 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! $\endgroup$ – User190212 Feb 14 at 0:31
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    $\begingroup$ The etiquette here is to ask a new question. Users look down on edits to questions that invalidate existing answers. $\endgroup$ – Martin Argerami Feb 14 at 0:34
  • $\begingroup$ Ok thanks, that was my intuition. Thank you again! $\endgroup$ – User190212 Feb 14 at 0:36
  • $\begingroup$ No problem :) $ $ $\endgroup$ – Martin Argerami Feb 14 at 0:52
  • $\begingroup$ What if $A$ is simple ? $\endgroup$ – Epsilon Feb 14 at 8:31

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