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

Round 1

Reviewer 1 Report

Add reference for Banach-Alaoglu theorem on p. 5.

Define the chi symbol is in Theorem 3.1. 

Comments for author File: Comments.pdf

Author Response

Many thanks for the suggestion.

I've prepared the new version with the following changes/improvemen;

-I added the reference for Banach Alaoglu theorem

-I added the definition of \chi

-I added a footnote about the regular representation of a discrete group (like the free group on countably many generators I used)

-corrected some misprints (and, I hope) I improved a bit the english exposition

Reviewer 2 Report

In the paper the author studies ergodic properties of non - commutative (quantum) dynamical systems, where algebras of functions are replaced by general C*  or W* - algebras, the action on functions of the transformation is replaced by positive map on the elements of the algebra and the counterpat of the invariant measure  is the invaiant state. The main goal of the paper is to provide non - commutative generalization of the "classical" result concerning the uniform convergence of Cesaro averages relative to the uniquelly ergodic dynamical systems. This goal is achieved in Theorem 3.1 which is proved using some preliminary results presented in Section 2.
The paper contains new and interesting results and is good written. Theefore I recommend this manuscript for publication.

Author Response

Many thanks for the suggestion.

I've prepared the new version with the following changes/improvemen;

-I added the reference for Banach Alaoglu theorem

-I added the definition of \chi

-I added a footnote about the regular representation of a discrete group (like the free group on countably many generators I used)

-corrected some misprints (and, I hope) I improved a bit the english exposition