Gradient forms and strong solidity of free quantum groups

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Abstract

Consider the free orthogonal quantum groups ON+(F) and free unitary quantum groups UN+(F) with N≥ 3. In the case F= id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra L∞(ON+) is strongly solid. Moreover, Isono obtains strong solidity also for L∞(UN+). In this paper we prove for general F∈ GLN(C) that the von Neumann algebras L∞(ON+(F)) and L∞(UN+(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.

Original languageEnglish
Pages (from-to)271–324
Number of pages54
JournalMathematische Annalen
Volume379 (2021)
Issue number1-2
DOIs
Publication statusPublished - 2020

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