Coarse rigidity in von Neumann algebras via derivations

Abstract

Motivated by the interaction between amenability and first cohomology into tempered representations of groups, we propose a notion of relative coarse rigidity for $\sigma$-finite von Neumann algebras through modular derivations and their associated GNS-symmetric quantum Markov semigroups. This extends Peterson's framework of $L^2$-rigidity to the nontracial setting. Our main result shows that the absence of relatively amenable summands forces the associated deformations to converge on relative commutants in ultrapowers. For diffuse von Neumann algebras with separable predual, this yields a dichotomy between the presence of an amenable summand and rigidity in terms of central sequences. Combining this principle with mixing and compactness, we provide a novel unified approach for proving indecomposability properties such as fullness, primeness and stable solidity. Applications yield new examples of relatively $\omega$-solid and relatively stably solid type III von Neumann algebras arising from amalgamated free products, operator-valued free Araki-Woods constructions, cocycle crossed products, and twisted nonsingular discrete measured groupoids.

arXiv commentv2: 92 pages. Minor updates to abstract and introduction. Added a new section 11 on primeness results and new examples on groupoid von Neumann algebras in Section 13. Updated acknowledgments, rearranged sections, and corrected minor typos