Orthogonal free quantum group factors are strongly 1-bounded

Brannan, Michael; Vergnioux, Roland

doi:10.1016/j.aim.2018.02.007

Cited by 14 publications

(19 citation statements)

References 39 publications

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“…[22]. However, for O + N , even more is known: In [23] it was shown that in fact L ∞ (O + N ) is a strongly 1-bounded von Neumann algebra for all N ≥ 3. The notion of strong 1-boundedness was introduced by Jung in [30] and entails that δ 0 (X) ≤ 1 for any self-adjoint generating set X ⊂ L ∞ (O + N ).…”

Section: Remarks On Free Entropy Dimensionmentioning

confidence: 99%

Topological generation and matrix models for quantum reflection groups

Brannan

Chirvăsitu

Freslon

2020

Advances in Mathematics

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We establish several new topological generation results for the quantum permutation groups S + N and the quantum reflection groups H s+ N . We use these results to show that these quantum groups admit sufficiently many "matrix models". In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain "flat" matrix models for S + N are inner faithful.

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Section: Remarks On Free Entropy Dimensionmentioning

confidence: 99%

Topological generation and matrix models for quantum reflection groups

Brannan

Chirvăsitu

Freslon

2020

Advances in Mathematics

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“…As has been known for over twenty years now, another source of operator algebras sharing many properties with these related to discrete groups is provided by the theory of compact (equivalently, discrete) quantum groups, as initiated by Woronowicz ([Wor98]), with the quantum theory encompassing its classical counterpart. Of particular interest is the class of universal quantum groups of Van Daele and Wang ([VDW96]), which leads to operator algebras behaving in many ways as these associated with the classical free groups ( [VV07]), but recently shown by Brannan and the last named author of this paper to be non-isomorphic to these at the von Neumann algebra level ( [BV18]).…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Furstenberg boundary

Kalantar,

Kasprzak,

Skalski

et al. 2020

Preprint

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We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C * -algebra admits at most one KMSstate for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.

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“…Ever since their introduction these algebras have received considerable attention and in particular over the last few years significant structural results have been obtained for them. In particular, recently it was proved that free quantum groups can be distinguished from the free group factors [11]. Further, the following is known if we assume N ≥ 3 (the case N = 2 corresponds to the amenable SU q (2) case):…”

Section: Introductionmentioning

confidence: 99%

Gradient forms and strong solidity of free quantum groups

Caspers

2020

Math. Ann.

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Consider the free orthogonal quantum groups $$O_N^+(F)$$ O N + ( F ) and free unitary quantum groups $$U_N^+(F)$$ U N + ( F ) with $$N \ge 3$$ N ≥ 3 . In the case $$F = \text {id}_N$$ F = id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $$L_\infty (O_N^+)$$ L ∞ ( O N + ) is strongly solid. Moreover, Isono obtains strong solidity also for $$L_\infty (U_N^+)$$ L ∞ ( U N + ) . In this paper we prove for general $$F \in GL_N(\mathbb {C})$$ F ∈ G L N ( C ) that the von Neumann algebras $$L_\infty (O_N^+(F))$$ L ∞ ( O N + ( F ) ) and $$L_\infty (U_N^+(F))$$ 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.

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Orthogonal free quantum group factors are strongly 1-bounded

Cited by 14 publications

References 39 publications

Topological generation and matrix models for quantum reflection groups

Topological generation and matrix models for quantum reflection groups

Noncommutative Furstenberg boundary

Gradient forms and strong solidity of free quantum groups

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