Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems - Francesco Fidaleo (Entropy, 2018)
Consider a uniquely ergodic <semantics> C * </semantics> -dynamical system based on a unital *-endomorphism <semantics> Φ </semantics> of a <semantics> C * </semantics> -algebra. We prove the uniform convergence of Cesaro averages <semantics> 1 n <msubsup> ∑ k = 0 n − 1 </msubsup> λ − n Φ ( a ) </semantics> for all values <semantics> λ </semantics> 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.