= Inversion reverses the order of strictly positive elements
{title2=$0<a\leq b\Longrightarrow0<b^{-1}\leq a^{-1}$}
Here strictly positive means positive and invertible. Set $k=a^{-1/2}ba^{-1/2}\geq1$ using <congruence preserves positivity in a C-star algebra>. The <continuous functional calculus> gives $k^{-1}\leq1$, since $1-t^{-1}\geq0$ for $t\geq1$. Then $a^{-1}-b^{-1}=a^{-1/2}(1-k^{-1})a^{-1/2}\geq0$. The proof works without commutativity. If $1\leq a$, functional calculus also gives $a^{-1}\leq1$.
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