= Positive cone of a C-star algebra
{title2=$A_+=\{h=h^*: \sigma(h)\subseteq[0,\infty)\}$}
The positive elements of a <C-star algebra> form a norm-closed proper convex cone. For $p,q\geq0$ and $s=\|p\|+\|q\|$, <continuous functional calculus> gives $\|s1-p-q\|\leq s$, forcing the real spectrum of $p+q$ to be nonnegative. The same bound with one fixed upper norm bound proves norm closedness under limits; scaling by nonnegative real scalars preserves positivity. If both $h$ and $-h$ are positive, their spectra lie in $\{0\}$, and <spectral radius norm equality for normal elements> forces $h=0$. This cone defines the order $a\leq b$ precisely when $b-a\in A_+$.
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