Furstenberg熵在ucp映射下的刚性
Rigidity of Furstenberg entropy under ucp maps
AI总结：
本文证明了迹von Neumann代数的平稳W$^*$-扩张间保持态的M-双模ucp映射保持Furstenberg熵当且仅当诱导Radon-Nikodym因子同构，将该结果推广至局部紧群拟因子映射并证明熵分离，最终为唯一平稳Poisson边界建立刚性现象。
摘要：
给定迹von Neumann代数$(M,τ)$，我们证明在$(M,τ)$的两个平稳W$^*$-扩张之间，保持态的$M$-双模ucp映射保持Furstenberg熵当且仅当它在Radon-Nikodym因子之间诱导同构。通过类似证明，我们将此结果推广到局部紧群的平稳空间间的拟因子映射，并证明唯一平稳空间与顺从空间间的熵分离。作为应用，我们利用这些结果为唯一平稳Poisson边界建立刚性现象。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
