Coarse rigidity in von Neumann algebras via derivations

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, extending 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. These results give a novel unified approach for proving indecomposability properties such as fullness, relative $\omega$-solidity and relative stable solidity. Our methods apply to examples such as amalgamated free products, operator-valued free Araki-Woods factors and cocycle crossed products.