Gradient forms and strong solidity of free quantum groups

TL;DR

This paper proves that von Neumann algebras associated with free quantum groups are strongly solid for general parameters, extending previous results and employing gradient bimodule analysis and derivation techniques.

Contribution

It establishes strong solidity for $L_ fty(O_N^+(F))$ and $L_ abla(U_N^+(F))$ for all invertible $F$, generalizing prior specific cases.

Findings

Strong solidity of $L_ fty(O_N^+(F))$ and $L_ abla(U_N^+(F))$ for all $F eq 0$

Analysis of gradient bimodules and Dirichlet forms on these algebras

Extension of previous results from identity to general $F$

Abstract

Consider the free orthogonal quantum groups $O_N^+(F)$ and free unitary quantum groups $U_N^+(F)$ with $N \geq 3$. In the case $F = {\rm id}_N$ 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.