Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries

Rijdt, An De; Vennet, Nikolas Vander

doi:10.5802/aif.2520

Cited by 54 publications

(82 citation statements)

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“…However, the converse may not hold since it is not clear that any torsion module of R G can be traced back to a torsion action of G. However, we will see that knowing the torsion modules of R G is often enough to understand the torsion actions of G. Let us conclude by a remark on the link between torsion actions and monoidal equivalence which will be crucial. If G and H are monoidally equivalent compact quantum groups in the sense of [6], then there is a one-to-one correspondence between their actions which was explicitly described in [8]. Moreover, this correspondence can be lifted to an equivalence of categories which preserves finitedimensionality and ergodicity as explained in [20,Sec 8].…”

Section: Definition 22mentioning

confidence: 99%

Torsion and K-theory for Some Free Wreath Products

Freslon

Martos

2018

International Mathematics Research Notices

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Section: Definition 22mentioning

confidence: 99%

Torsion and K-theory for Some Free Wreath Products

Freslon

Martos

2018

International Mathematics Research Notices

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“…Moreover, in [11], the authors prove that the same construction with the inverse monoidal equivalence ϕ −1 will give D 1 again up to isomorphism. Theorem 1.17.…”

Section: Monoidal Equivalences Between Compact Quantum Groupsmentioning

confidence: 95%

“…There is even more, De Rijdt and Vander Vennet proved in [11] that there exists a bijection between actions of monoidal equivalent compact quantum groups. Indeed, let G 1 and G 2 be two compact quantum groups, ϕ : G 1 → G 2 be a monoidal equivalence between them.…”

Section: Monoidal Equivalences Between Compact Quantum Groupsmentioning

confidence: 99%

Deformations of spectral triples and their quantum isometry groups via monoidal equivalences

Sadeleer

2016

Lett Math Phys

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“…Remark 9.3. It is also worth mentioning that Theorem 9.1 has also been used together with monoidal equivalence and free product results from [36] and [35] to obtain analogous results for the discrete duals of quantum automorphism groups of finite dimensional C˚-algebras.…”

Section: Application: Approximation Properties For Free Quantum Groupsmentioning

confidence: 96%

Approximation properties for locally compact quantum groups

Brannan¹

2017

Banach Center Publ.

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This is a survey of some aspects of the subject of approximation properties for locally compact quantum groups, based on lectures given at the Topological Quantum Groups Graduate School, 28 June -11 July, 2015 in Bed lewo, Poland. We begin with a study of the dual notions of amenability and co-amenability, and then consider weakenings of these properties in the form of the Haagerup property and weak amenability. For discrete quantum groups, the interaction between these properties and various operator algebra approximation properties are investigated. We also study the connection between central approximation properties for discrete quantum groups and monoidal equivalence for their compact duals. We finish by discussing the central weak amenability and central Haagerup property for free quantum groups.Date: May 9, 2016. 1 2 there have been very recent and striking developments on questions related to residual finiteness, (inner)linearity, hyperlinearity, and the Kirchberg factorization property for discrete quantum groups. See [5,6,14,23,9].Acknowledgements. First and foremost, I would like to thank Uwe Franz, Adam Skalski, and Piotr Soltan for organizing such a wonderful graduate school, for inviting me to speak there, and for encouraging me to write this survey. I would also like to thankÉtienne Blanchard and Stefaan Vaes for kindly sharing with me their unpublished notes [11]. PreliminariesIn the following, we write b for the minimal tensor product of C˚-algebras or tensor product of Hilbert spaces, b for the spatial tensor product of von Neumann algebras and d for the algebraic tensor product. All inner products are taken to be conjugate-linear in the second variable. Given vectors ξ, η in a Hilbert space H, we denote by ω ξ,η P BpHq˚the vector state x Þ Ñ xxξ|ηy. If ξ " η, we simply write ω ξ " ω ξ,ξ . 2.1. Locally compact quantum groups. Let us first recall from [55] and [56] that a (von Neumann algebraic) locally compact quantum group (lcqg) is a quadruple G " pM, ∆, ϕ, ψq, where M is a von Neumann algebra, ∆ : M Ñ MbM is a co-associative coproduct, i.e. a unital normal˚-homomorphism such that pι b ∆q˝∆ " p∆ b ιq˝∆, and ϕ and ψ are normal faithful semifinite weights on M such that pι b ϕq∆pxq " ϕpxq1 and pψ b ιq∆pxq " ψpxq1 px P M`q.We call ϕ and ψ the left Haar weight and the right Haar weight of G, respectively, and we write L 8 pGq for the quantum group von Neumann algebra M. Associated with each locally compact quantum group G, there is a reduced quantum group C*-algebra C 0 pGq Ď L 8 pGq with coproduct Let L 1 pGq " L 8 pGq˚be the predual of L 8 pGq. Then the pre-adjoint of ∆ induces on L 1 pGq a completely contractive Banach algebra productwhere we let p b denote the operator space projective tensor product. We will also use the symbol ‹ to denote the induced left/right module actions of L 1 pGq on L 8 pGq:Using the Haar weights, one can construct an antipode S on L 8 pGq satisfying pS b ιqW " W˚. Since, in general, S is unbounded on L 8 pGq, we can not use S to define an involution on L 1 pGq. Howev...

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Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries

Cited by 54 publications

References 20 publications

Torsion and K-theory for Some Free Wreath Products

Torsion and K-theory for Some Free Wreath Products

Deformations of spectral triples and their quantum isometry groups via monoidal equivalences

Approximation properties for locally compact quantum groups

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