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 = text {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 GL_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.