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

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

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