Gradient forms and strong solidity of free quantum groups
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Mathematische Annalen
Gradient forms and strong solidity of free quantum groups Martijn Caspers1 Received: 12 February 2018 / Revised: 23 April 2020 / Accepted: 20 October 2020 © The Author(s) 2020
Abstract Consider the free orthogonal quantum groups O N+ (F) and free unitary quantum groups U N+ (F) with N ≥ 3. In the case F = id N it was proved both by Isono and FimaVergnioux that the associated finite von Neumann algebra L ∞ (O N+ ) is strongly solid. Moreover, Isono obtains strong solidity also for L ∞ (U N+ ) . In this paper we prove for general F ∈ G L N (C) that the von Neumann algebras L ∞ (O N+ (F)) and L ∞ (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.
1 Introduction In their fundamental paper [47] Ozawa and Popa gave a new method to show that the free group factors do not possess a Cartan subalgebra, a result that was obtained earlier by Voiculescu [66] using free entropy. To achieve this, Ozawa and Popa in fact proved a stronger result. They showed that the normalizer of any diffuse amenable von Neumann subalgebra of the free group factors, generates a von Neumann algebra that is again amenable. This property then became known as ‘strong solidity’. As free group factors are non-amenable and strongly solid they in particular cannot contain Cartan subalgebras. The approach of [47] splits into two important parts. The first is the notion of ‘weak compactness’. [47] showed that if a von Neumann algebra has the CMAP, then the normalizer of an amenable von Neumann subalgebra acts by conjugation on the subalgebra in a weakly compact way. The second part consists in combining weak compactness with Popa’s malleable deformation for the free groups and his spectral gap techniques.
Communicated by Andreas Thom.
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Martijn Caspers [email protected] TU Delft, EWI/DIAM, P.O. Box 5031, 2600 GA Delft, The Netherlands
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M. Caspers
After the results of Ozwa-Popa several other strong solidity results have been obtained by combining weak compactness with different deformation techniques of (group-) von Neumann algebras, often coming from group geometric properties. Roughly (to the knowledge of the author) they can be divided into three categories: (I.1) The aforementioned malleble deformations; (I.2) The existence of proper cocycles and derivations and deformations introduced by Peterson [50] and further developed by Ozawa–Popa [48]; (I.3) The Akemann-Ostrand property, which compares to proper quasi-cocycles and bi-exactness of groups; c.f. [12,19,54]. For group von Neumann algebras the required property in (I.2) is to a certain extent stronger than (I.3) in the sense that proper cocycles are in particular quasi-cocycles. These techniques have been applied successfully to obtain rigidity results for von Neumann algebras (in particular strong solidity results). The current paper also obtains such results and our global methods fall
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