NON-COMMUTATIVE SKEW-PRODUCT EXTENSION DYNAMICAL SYSTEMS

The paper proves the equivalence among (i) unique ergodicity w.r.t. the fixed-point subalgebra, (ii) uniqueness of a G-invariant conditional expectation onto the fixed-point subalgebra, and (iii) the absence of measurable-but-noncontinuous coboundaries for all derived cocycles u^(n) in the skew-product setting. The candidate solution reaches the same equivalence via a similar chain, correctly using Proposition 2.2 and (ii)⇒(iii) ideas, and appealing to (iii)⇒(i) in the skew-product context. However, its Step 3 claims (ii)⇒(i) by invoking norm convergence of Følner averages from the mere uniqueness of the invariant conditional expectation; this implication is not valid in general (Ursu’s counterexample), though the full equivalence still follows in the skew-product class from the model’s remaining steps. Overall, the paper is correct and complete, and the model is essentially correct modulo one unnecessary and flawed inference.