Rigidity of Furstenberg entropy under ucp maps
Shuoxing Zhou
TL;DR
The paper addresses rigidity phenomena for Furstenberg entropy under state-preserving $M$-bimodular ucp maps between stationary W$^*$-extensions and, more generally, quasi-factor maps between stationary group spaces. It develops a framework based on noncommutative Radon–Nikodym factors to compare entropy, proving a monotonicity result $h_\varphi(\mathcal{A},\varphi_\mathcal{A})\le h_\varphi(\mathcal{B},\varphi_\mathcal{B})$ with equality iff the RN-factor restriction is a $\ast$-isomorphism; it also extends these ideas to quasi-factor maps and group actions. The results yield entropy separation between unique stationary and amenable spaces and lead to rigidity statements for unique stationary Poisson boundaries, including RN-irreducibility characterizations and equivalences involving amenability. Consequently, the work provides new proofs and unifications of several results in both noncommutative and ergodic group theory (e.g., HK24, FG10, NS13, Hou24) by an entropy-centric, RN-factor perspective.
Abstract
Given a tracial von Neumann algebra $(M,τ)$, we prove that a state preserving $M$-bimodular ucp map between two stationary W$^*$-extensions of $(M,τ)$ preserves the Furstenberg entropy if and only if it induces an isomorphism between the Radon-Nikodym factors. With a similar proof, we extend this result to quasi-factor maps between stationary spaces of locally compact groups and prove an entropy separation between unique stationary and amenable spaces. As applications, we use these results to establish rigidity phenomena for unique stationary Poisson boundaries.