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

The functional's stipulated positivity initially means $f(z^{\ast }z)\ge 0$, without assuming continuity. For a Hermitian element of a C-star algebra $h$, the identityexpresses it as a real linear combination of two adjoint squares. Their functional values are real, so $f(h)$ is real. Write $x=h+ik$ with $h=(x+x^{\ast })/2$ and $k=(x-x^{\ast })/(2i)$ Hermitian. Complex linearity gives $f(x)=f(h)+if(k)$ and $f(x^{\ast })=f(h)-if(k)$. ThereforeThis is the Hermitian symmetry used in the Cauchy–Schwarz inequality for positive C-star functionals.