Proposition 41.2.4.label Let $A$ be a $C^{*}$-algebra, $S \subset A$ be an operator system, and $x \in S_{sa}$, then there exists positive elements $p, q \in S_{sa}$ such that $x = p - q$.
Proof. Let $p = (\norm{x}_{A} + x)/2$ and $q = (\norm{x}_{A} - x)/2$, then $q$ is positive by Corollary 38.6.3.$\square$
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