Noncommutative Ergodic Theory of Higher Rank Lattices

The paper’s theorem (Theorem E) states exactly the desired dichotomy and provides a sound proof sketch (via induction to a G-boundary structure, construction of a suitable subalgebra, passage to a commutative center, and evaluation back to the lattice), with details deferred to prior work. By contrast, the model’s proof hinges on a multiplicative-domain minimal-norm argument that assumes, without justification, that for every f in L∞(G/Q) the affine constraint set {x in M : Ψ(x)=f} is nonempty; this is precisely the surjectivity it seeks to prove. The step from “the von Neumann algebra generated by Ψ(M) equals L∞(G/Q)” to “Ψ(M)=L∞(G/Q)” is unjustified. Hence the model proof has a critical gap, while the paper’s argument is consistent with the literature.