I have the following automorphsim of $\displaystyle C(\mathbb{T})$
$\displaystyle \tau_{\theta}(f)(z)=f(e^{2\pi i \theta}z)=(R^{-1}_{\theta}\circ f)(z)$
with $\displaystyle R_{\theta}$ rotation the argument of the functions $\displaystyle f$ by $\displaystyle 2\pi i\theta$

Let $\displaystyle Z$ be the image of the coordinate function $\displaystyle z$ and let $\displaystyle W$ be a unitary implementing the above mentioned automorphism $\displaystyle \tau_{\theta}$. (What exactly is meant by implementing an automorphism? Is it something like $\displaystyle W(f)=\tau_{\theta}(f)$ where $\displaystyle f$ then is some unitary function.)

show that
$\displaystyle WZW^*=\tau_{\theta}(z)(Z)=e^{-2\pi i\theta}Z$