COAMENABILITY AND COSPECTRAL RADIUS FOR ORBIT EQUIVALENCE RELATIONS

The paper’s Theorem 1.1 proves the equivalence of five characterizations of coamenability for S ≤ R (ergodic), namely (i) hypertrace on ⟨L(R),e_{L(S)}⟩, (ii) almost invariant vectors for λ_{R/S}, (iii) Kesten-type norm one for every symmetric generating ν on [R], (iv) fiberwise Reiter-type almost invariance, and (v) the same for a fixed countable Γ ≤ [R] with Γx=[x]_R a.e. This matches the model’s target and scope. The paper establishes the operator-norm/ cospectral-radius identity and the circle of implications via an invariant-mean framework and a careful operator-algebraic link to the basic construction, culminating in the vNA equivalence L(S) coamenable in L(R) (Theorem 6.11) and a broad equivalence theorem (Theorem 3.5) that includes (ii)–(v) and the norm-one condition (iii) . The model’s solution proves the same equivalences but routes (i)⇔(ii)⇔(iv) through Anantharaman-Delaroche’s relative amenability, and (ii)⇔(iii) via a representation-theoretic spectral-gap contrapositive (Bekka–Guivarc’h) after exploiting the paper’s norm identity. These are valid alternative pathways. Minor imprecisions in the model (e.g., an unnecessary claim of uniform-in-fiber convergence and an over-strong statement that certain vector states equal the canonical trace rather than converge to it) do not affect correctness. Overall, both are correct; the proofs are substantially different in technique.