The radial MASA in free orthogonal quantum groups

Freslon, Amaury; Vergnioux, Roland

doi:10.1016/j.jfa.2016.08.007

Cited by 7 publications

(5 citation statements)

References 20 publications

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“…Using the next lemma we will obtain a better estimate in the proof of Theorem 7.13 -in the Kac case even a much better one since we will deduce that w k,l n,t (ξ) HS is bounded with respect to n, for fixed k. In the proof we use the "higher weight" Wenzl recursion relation [FV16,Lemma 3. (2) ⊗ id n−p−q )P + n−q , where the coefficient α n p,q satisfies |α n p,q | ≤ 1. This formula is stated in [FV16] only in the Kac case -since the methods used in that article to study MASAs only hold in the tracial case. However a rapid inspection shows that its statement and proof hold without modification in the general, non-Kac case.…”

Section: Boundary Actions For Free Orthogonal Quantum Groupsmentioning

confidence: 94%

“…We denote κ r,s t the inverse of its norm, so that V t r,s = κ r,s t (P + r ⊗ P + s )(id ⊗ t a ⊗ id)P + t is an isometric intertwiner. The quantity κ r,s t has been studied by many authors: here we follow the notation of [FV16], in [BC18] it is denoted κ r,s t = A r,s t −1 = [t + 1] q θ q (t, r, s)…”

Section: Boundary Actions For Free Orthogonal Quantum Groupsmentioning

confidence: 99%

“…This will go into the third term on the right-hand side of (7.5). Now we use [FV16,Lemma 3.4]. We want to apply it to σ(ξ) := F −1 l ξ (2) ⊗ ξ (1) which is not in H k (even if ξ was), so we first decompose: σ(ξ) = ξ ′ + (P + l ⊗ P + k−l )(id l−1 ⊗ t 1 ⊗ id k−l−1 )(ξ ′′ 0 )…”

Section: Hence We Havementioning

confidence: 99%

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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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Section: Boundary Actions For Free Orthogonal Quantum Groupsmentioning

confidence: 94%

Section: Boundary Actions For Free Orthogonal Quantum Groupsmentioning

confidence: 99%

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Noncommutative Furstenberg boundary

Kalantar,

Kasprzak,

Skalski

et al. 2020

Preprint

Self Cite

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“…This question was studied in particular in the case of the (Kac type) free orthogonal quantum group O + N . In [FV16] We choose to call C G "the von Neumann algebra of class functions" because the two coincide for classical compact groups (see Lemma 1.2).…”

Section: Introductionmentioning

confidence: 99%

On the von Neumann algebra of class functions on a compact quantum group

Krajczok¹,

Wasilewski²

2021

Preprint

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We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are in a non-Kac situation. The most important notion to our present work is that of a (quasi-)split inclusion. We prove that the inclusion of the algebra of class functions is quasi-split for some unitary quantum groups -in this case the subalgebra is non-abelian and we also obtain a result concerning its relative commutant. In the positive direction, we construct certain bicrossed products from the quantum group SUq(2) for which the algebra of class functions is a MASA. 2020 Mathematics Subject Classification. Primary 46L67, 20G42. Key words and phrases. Compact quantum group, Quasi-split inclusion, Maximal abelian subalgebra. 1 Note however that for G = Fn we do not have equality of R and C G . In fact, C G = L ∞ (G) holds for all abelian compact quantum groups.

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“…The study of masas and their bimodule‐related properties in the context of CQGs is rather recent and was carried out in the context of orthogonal quantum groups by Freslon and Vergnioux in . Our line of investigation of CQG dynamical systems finds applications in this direction; thus we study masas and their bimodules in CQGs which arise from group inclusions.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of compact quantum groups

Mukherjee

Patri

2017

Proc. London Math. Soc.

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This paper initiates the study of non-commutative dynamical systems of the form (G, Γ), where a discrete group Γ acts on a compact quantum group (CQG) G by quantum automorphisms. We obtain combinatorial conditions for such dynamical systems to be ergodic, mixing, compact, etc. and provide a wide variety of examples to illustrate these conditions. We generalize a well-known theorem of Halmos to demonstrate 'reversal of arrows' in the ergodic hierarchy relevant to the context and make a study of spectral measures for actions of (non-commutative) groups. We investigate the structure of such dynamical systems and under certain restrictions exhibit the existence and uniqueness of the maximal ergodic invariant normal subgroup of such systems. As an application, we study the size of normalizing algebras of masas arising from groups in von Neumann algebraic CQGs and show that the normalizing algebra of such masas are the von Neumann algebras generated by co-commutative CQGs.

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The radial MASA in free orthogonal quantum groups

Cited by 7 publications

References 20 publications

Noncommutative Furstenberg boundary

Noncommutative Furstenberg boundary

On the von Neumann algebra of class functions on a compact quantum group

Automorphisms of compact quantum groups

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