On uniform ergodic theorems for quadratic processes on $C^*$-algebras

We investigate questions of the convergence in the uniform topology of Ceséaro time averages of stationary quantum quadratic processes defined on $C^*$-algebras.
A necessary and sufficient condition is given for the convergence of averages in the uniform topology. In addition, the convergence of weighted averages in the uniform topology is investigated and it is proved that averages weighted by Besicovitch $\Phi$-functions converge.