On a function that realizes the maximal spectral type

We show that for a unitary operator U on ${\displaystyle L^2(X,\mu )}$, where X is a compact manifold of class ${\displaystyle C^r}$, ${\displaystyle r\in \mathbb{N}\cup \{\infty ,\omega \}}$, and μ is a finite Borel measure on X, there exists a ${\displaystyle C^r}$ function that realizes the maximal spectral type of U.