Uniform Convergence of Cesaro Averages for Uniquely Ergodic C(*)-Dynamical Systems

Consider a uniquely ergodic [Formula: see text]-dynamical system based on a unital *-endomorphism [Formula: see text] of a [Formula: see text]-algebra. We prove the uniform convergence of Cesaro averages [Formula: see text] for all values [Formula: see text] in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.