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This is a bit of an open-ended question... Let $S$ be an operator algebra (or an operator system) and consider a functional $\nu:M\to \mathbb{C}$ that satisfies $$\vert \nu(a)\vert \ge -\varepsilon \Vert a\Vert$$ for every $a\ge A$? My question is:

Is there a positive functional near $f$? By this I mean is there $\delta>0$ the relies on $\varepsilon$ and a positive functional $\mu:M\to \mathbb{C}$ such that $\Vert \mu-\nu\Vert\le \delta$?

If the answer is no in general, are there any conditions that would imply a positive answer? Or would it work for some examples maybe?

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    $\begingroup$ Could you clarify the meanings of $S$, $A$ and $M$ in your statement? $\endgroup$
    – Ruy
    Commented Apr 24, 2017 at 0:58

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