I am reading Sec33 of Conway's A Course in Operator Theory, according to his definition,
An operator system is a linear manifold $\mathcal{S}$ in a $C^*$-algebra such that $1\in\mathcal{S}$ and $\mathcal{S}=\mathcal{S}^*$.
Then he makes the comment that operator systems have an abundance of positive elements. His argument is that for every hermitian element $a\in\mathcal{S}$, $|a|$ also lies in $\mathcal{S}$ and hence the positive parts and negative parts lie in $\mathcal{S}$ and hence $\mathcal{S}$ is spanned by its positive elements.
However, I do not know why $|a|$ lies in $\mathcal{S}$ since $\mathcal{S}$ is only assumed to be a linear manifold, not an algebra.
Can somebody give a hint? Thanks!