= Solution
The functional's stipulated positivity initially means $f(z^*z)\geq0$, without assuming continuity. For a <Hermitian element of a C-star algebra> $h$, the identity
$$
h=\tfrac14\bigl((h+1)^2-(h-1)^2\bigr)
$$
expresses 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 $h=(x+x^*)/2$ and $k=(x-x^*)/(2i)$ Hermitian. Complex linearity gives $f(x)=f(h)+if(k)$ and $f(x^*)=f(h)-if(k)$. Therefore
$$
\boxed{f(x^*)=\overline{f(x)}.}
$$
This is the Hermitian symmetry used in the <Cauchy–Schwarz inequality for positive C-star functionals>.
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