= Cauchy–Schwarz inequality for positive C-star functionals
{c}
{title2=$|f(x^*y)|^2\leq f(x^*x)f(y^*y)$}
Positivity of $f((x+ty)^*(x+ty))$ for every complex $t$ proves the displayed <Cauchy-Schwarz inequality>, including zero diagonal cases. Replacing $x,y$ by their adjoints gives $|f(xy^*)|^2\leq f(xx^*)f(yy^*)$. These two forms cannot be mixed for a general <positive functional on a C-star algebra>. For the state evaluating the lower-right entry of $2\times2$ matrices and $x=y=E_{12}$, $f(x^*y)=1$ but $f(xx^*)=f(yy^*)=0$, disproving the mixed form.
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