W*-superrigidity for discrete quantum groups
Milan Donvil, Stefaan Vaes
TL;DR
The paper develops a quantum analogue of W*-superrigidity for discrete quantum groups by combining three key ingredients: (i) a detailed analysis of unitary 2-cocycles and their obstructions to quantum W*-superrigidity, (ii) notions of relative rigidity that recover the quantum group structure from von Neumann data under group actions, and (iii) a coarse-embedding classification for co-induced Bernoulli crossed products that enables Popa-style deformation/rigidity arguments. The authors construct co-induced left-right Bernoulli crossed products from a wide range of actions and show these yield quantum W*-superrigid compact quantum groups, while also identifying broad families of classical W*-rigid groups that fail to be quantum W*-rigid. They also provide extensive criteria (including vanishing 2-cohomology) under which cocycle twists cannot occur, ensuring rigidity in many natural examples such as connected abelian groups, certain finite groups like SL_n(𝔽_q) and their semidirect products, and specific covers like ˜A_n. Overall, the work demonstrates a sharp separation between W*-rigidity for discrete groups and quantum W*-rigidity, and it provides a robust framework for constructing and certifying quantum W*-superrigidity via cocycle vanishing, relative rigidity, and coarse-embedding techniques.
Abstract
A discrete group $G$ is called W*-superrigid if the group $G$ can be entirely recovered from the ambient group von Neumann algebra $L(G)$. We introduce an analogous notion for discrete quantum groups. We prove that this strengthened quantum W*-superrigidity property holds for a natural family of co-induced discrete quantum groups. We also prove that, remarkably, most existing families of W*-superrigid groups are not quantum W*-superrigid.