scholarly journals Beurling-Fourier algebras f compact quantum groups: characters and finite dimensional representations

2021 ◽  
Vol 70 (2) ◽  
pp. 605-637
Author(s):  
Uwe Franz ◽  
Hun Hee Lee
Keyword(s):  
Quantum Groups ◽  
Finite Dimensional ◽  
Compact Quantum Groups

scholarly journals Representations of compact quantum groups and subfactors

1999 ◽  
Vol 1999 (509) ◽  
pp. 167-198 ◽  
Author(s):  
Teodor Banica
Keyword(s):  
Quantum Groups ◽  
Hilbert Spaces ◽  
Type Result ◽  
Free Products ◽  
Finite Dimensional ◽  
Compact Quantum Groups ◽  
Universal Construction

Abstract We associate Popa systems (= standard invariants of subfactors) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented” on finite dimensional Hilbert spaces. This is proved by a universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and the back via the universal construction. We prove a Kesten type result for the co-amenability of compact quantum groups, which allows us to compare it with the amenability of subfactors.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

scholarly journals Weak mixing for locally compact quantum groups

2016 ◽  
Vol 37 (5) ◽  
pp. 1657-1680 ◽  
Author(s):  
AMI VISELTER
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebras ◽  
Unitary Representations ◽  
Discrete Groups ◽  
Weak Mixing ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Weakly Mixing

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

scholarly journals The canonical central exact sequence for locally compact quantum groups

2016 ◽  
Vol 290 (8-9) ◽  
pp. 1303-1316 ◽  
Author(s):  
Paweł Kasprzak ◽  
Adam Skalski ◽  
Piotr Mikołaj Sołtan
Keyword(s):  
Exact Sequence ◽  
Quantum Groups ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups
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