2110.01386
TOPOLOGICAL GROUP ACTIONS BY GROUP AUTOMORPHISMS AND BANACH REPRESENTATIONS
Michael Megrelishvili
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves (i) GL(n,R) acting on R^n is not Asplund-representable and (ii) is Rosenthal-representable, via HNS/weak mixing and an enveloping-semigroup cardinality bound, respectively. The candidate’s part (b) matches the paper’s Rosenthal/tameness argument. However, in part (a) the candidate incorrectly asserts that the GL(n,R)-action on all of R^n is weakly mixing and misapplies the diagonal-action criterion; this step is essential for their contradiction and is false as stated. The paper instead constructs a weakly mixing subspace and invokes HNS, avoiding this error.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
Conceptually solid and technically correct results distinguishing Asplund vs. Rosenthal representability for classical matrix-group actions. The HNS obstruction and tameness characterization are deployed effectively, with clear exemplars. Some wording around the weakly mixing subspace in the Asplund obstruction could be clarified, but this does not affect correctness. The paper is a useful contribution for specialists studying Banach representations of group actions.