Partial C*-Dynamics and Rokhlin Dimension

The paper proves: (i) finite Rokhlin dimension implies freeness (Proposition 5.2), and (ii) for free topological partial actions on finite-dimensional spaces, the Rokhlin dimension is finite with an explicit bound via a decomposition-and-extension argument (Theorem 5.10). The model’s Part (a) incorrectly infers near-constancy of Rokhlin tower values across all group elements from condition (1) of Definition 2.1; that inference only propagates along cosets of the subgroup generated by the chosen element, so the subsequent uniform smallness conclusion is invalid. The paper’s correct proof avoids this pitfall and obtains a uniform pointwise bound f_h(x)<√(2ε) directly (). In Part (b), the model’s direct construction passes to an ad hoc orbit-space quotient and assumes partitions of unity and a finite-to-one dimension bound without justifying Hausdorffness/normality or closedness of the equivalence relation; these steps are precisely the delicate points the paper resolves via decompositions and lifting lemmas (). The model also claims the stronger bound dimRok(α) ≤ dim(X), which is not supported by the paper’s general result, where the stated bound is of the form (|G|−1)(dim(X)+1) ().