Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 10 / 5 / ii / Solution

The inequality printed in this part is false for general positive functionals. In $A=M_2(\mathbb{C})$ take the state on a C-star algebra $f(a)=a_{22}$ and $x=y=E_{12}$. It is positive since $f(z^{\ast }z)=\Vert z{e}_2{\Vert }^2\ge 0$, and it has norm one. ButThus the literal requested inequality would say $1\le 0$.

The correct Cauchy–Schwarz inequality for positive C-star functionals keeps the orders consistent:To prove it, set $a=f(x^{\ast }x)$, $d=f(y^{\ast }y)$ and $b=f(x^{\ast }y)$. By part (i) and positivity,If $d>0$, choose $t=-\overline{b}/d$ to obtain $|b{|}^2\le ad$. If $d=0$, arbitrary choices of $t$ force $b=0$, and the inequality again follows. Replacing $x,y$ by $x^{\ast },y^{\ast }$ also yields the valid alternativeEither replacing the printed right-hand factors by $f(x^{\ast }x),f(y^{\ast }y)$, or replacing the left-hand product by $x{y}^{\ast }$, repairs the statement. No tracial hypothesis is supplied, so it cannot be silently used to exchange those products.