= Squaring is not order preserving in a C-star algebra
{title2=$1\leq a\leq b\not\Longrightarrow a^2\leq b^2$}
In $M_2(\mathbb C)$ take $a=\operatorname{diag}(1,2)$ and $b=\begin{pmatrix}2&1\\1&3\end{pmatrix}$. Then $a-1\geq0$ and $b-a=\begin{pmatrix}1&1\\1&1\end{pmatrix}\geq0$, but $b^2-a^2=\begin{pmatrix}4&5\\5&6\end{pmatrix}$ has <determinant> $-1$ and a negative <eigenvalue>. In contrast, for commuting positive elements $b^2-a^2=(b-a)(b+a)$ is positive in the commutative function algebra supplied by <continuous functional calculus>. The obstruction is noncommutativity, not a failure of the scalar inequality.
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