## unitary implementing an automorphism

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