Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

Shuoxing Zhou, Tattwamasi Amrutam (appendix), Yongle Jiang (appendix)

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that any amenable Γ-invariant intermediate subalgebra A ⊂ N ⊂ Γ ⋉ A embeds in Rad(Γ) ⋉ A by adjoining a (commutative) Γ-boundary inside B(ℓ2Γ) ⊗ A, invoking the M-boundary property to obtain an M-bimodular completely isometric u.c.p. map into JN′J, using a multiplicative-domain argument to push the boundary copy into ℓ∞(Γ) ⊗ 1, and then identifying the basic construction to conclude N ⊂ Rad(Γ) ⋉ A. The candidate solution follows the same strategy with essentially the same ingredients and conclusions. Minor issues are (i) a mis-citation of theorem numbering and (ii) a technical slip writing E ⊗ idA instead of the faithful conditional expectation E ⊗ τA used in the paper, and (iii) a notational clash around ρg. None affect the core argument.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript develops a noncommutative boundary framework and leverages it to control amenable Γ-invariant (random) intermediate subalgebras in crossed products. The arguments are correct and well-motivated, and the results sharpen and generalize existing work. Some technical steps (conditional expectations and invariance actions) could be stated a touch more explicitly to aid readability, but the overall presentation is solid.