Cauchy–Schwarz inequality for positive C-star functionals ($|f(x^{\ast }y){|}^2\le f(x^{\ast }x)f(y^{\ast }y)$)

Positivity of $f((x+ty{)}^{\ast }(x+ty))$ for every complex $t$ proves the displayed Cauchy-Schwarz inequality, including zero diagonal cases. Replacing $x,y$ by their adjoints gives $|f(x{y}^{\ast }){|}^2\le f(x{x}^{\ast })f(y{y}^{\ast })$. 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 $2\times 2$ matrices and $x=y=E_{12}$, $f(x^{\ast }y)=1$ but $f(x{x}^{\ast })=f(y{y}^{\ast })=0$, disproving the mixed form.