Preprint Article Version 1 This version is not peer-reviewed

# Uniform Convergence of Cesaro Averages for Uniquely Ergodic $C^*$-Dynamical Systems

Version 1 : Received: 20 November 2018 / Approved: 22 November 2018 / Online: 22 November 2018 (04:17:04 CET)

A peer-reviewed article of this Preprint also exists.

Fidaleo, F. Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems. Entropy 2018, 20, 987. Fidaleo, F. Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems. Entropy 2018, 20, 987.

Journal reference: Entropy 2018, 20, 987
DOI: 10.3390/e20120987

## Abstract

Consider a uniquely ergodic $C^*$-dynamical system ba\-sed on a unital $*$-endomorphism $\Phi$ of a $C^*$-algebra. We prove the uniform convergence of Cesaro averages $\frac1{n}\sum_{k=0}^{n-1}\lambda^{-n}\Phi(a)$ for all values $\lambda$ in the unit circle which are not eigenvalues corresponding to "measurable non continuous" eigenfunctions. This result generalises the analogous one in commutative ergodic theory presented in [19], which turns out to be a combination of the Wiener-Wintner Theorem (cf. [22]) and the uniformly convergent ergodic theorem of Krylov and Bogolioubov (cf. [15]).

## Subject Areas

ergodic theorems; C*-dynamical systems