= Positivity of adjoint products from spectral positivity
{title2=$a^*a\geq0$}
Define positivity spectrally: a self-adjoint element has nonnegative spectrum. First, addition preserves it. For $p,q\geq0$ and $s=\|p\|+\|q\|$, functional calculus gives $\|s1-p-q\|\leq s$, forcing every real spectral value of $p+q$ to be nonnegative. The cone is proper because a self-adjoint element with spectrum contained in $\{0\}$ has norm zero.
For $h=a^*a$, let $h_-=\max(-h,0)$ by the <continuous functional calculus>, and set $c=ah_-^{1/2}$. Then $c^*c=-h_-^2$ is nonpositive. The <nonzero spectra of products in opposite orders> imply that $cc^*$ is nonpositive too. Write $c=u+iv$ with $u,v$ self-adjoint. Then $c^*c+cc^*=2(u^2+v^2)$ is positive, since self-adjoint squares have nonnegative spectrum. It is also nonpositive, hence zero. It follows that $c^*c=-cc^*$ is both positive and nonpositive, hence zero. The <C-star identity> gives $c=0$, so $h_-^2=0$ and $h_-=0$. Thus $h\geq0$, without assuming positivity of adjoint products in an earlier step.
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