Gradient forms and strong solidity of free quantum groups

Abstract

Consider the free orthogonal quantum groups $O_N^{+}(F)$ and free unitary quantum groups $U_N^{+}(F)$ with $N\ge 3$. In the case $F={\mathrm{id}}_N$ it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $L_{\infty }(O_N^{+})$ is strongly solid. Moreover, Isono obtains strong solidity also for $L_{\infty }(U_N^{+})$ . In this paper we prove for general $F\in G{L}_N(\mathbb{C})$ that the von Neumann algebras $L_{\infty }(O_N^{+}(F))$ and $L_{\infty }(U_N^{+}(F))$ are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani--Sauvageot.

Accepted for Mathematische Annalen