Congruence preserves positivity in a C-star algebra
= Congruence preserves positivity in a C-star algebra
{title2=$a\geq0\Longrightarrow c^*ac\geq0$}
For a positive element $a$, its <positive square root in a C-star algebra> gives $c^*ac=(a^{1/2}c)^*(a^{1/2}c)$, which is positive. Applying this to $b-a$ shows that $a\leq b$ implies $c^*ac\leq c^*bc$. The map need not preserve multiplication, but it preserves order; if $c$ is invertible, the inverse congruence also preserves order.