Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 10 / 3 / d / Solution

Write $\varphi (g)=a+ib$ with $a,b\in \mathbb{R}$, where $g$ is real valued. The normalization and the operator norm bound imply, for every real $t$,The left side is $(1-tb{)}^2+t^2{a}^2$, soDivide by $t>0$ and let $t\downarrow 0$ to obtain $b\ge 0$. Divide by $t<0$, reversing the inequality, and let $t\uparrow 0$ to obtain $b\le 0$. Thus $\boxed{\mathrm{Im}\varphi (g)=0}$ for every real-valued $g$. The first-order term in $t$ is what forces reality; this is the complex-functional step in the unital contraction positivity criterion.