scholarly journals Inner amenability and approximation properties of locally compact quantum groups

2019 ◽  
Vol 68 (6) ◽  
pp. 1721-1766 ◽  
Author(s):  
Jason Crann
Keyword(s):  
Quantum Groups ◽  
Approximation Properties ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Inner Amenability

Ergodic properties and harmonic functionals on locally compact quantum groups

2014 ◽  
Vol 25 (05) ◽  
pp. 1450051 ◽  
Author(s):  
Mehdi Nemati
Keyword(s):  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Ergodic Properties ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Inner Amenability

For a locally compact quantum group 𝔾, we generalize some notions of amenability such as amenability of locally compact quantum groups and inner amenability of locally compact groups to the case of right Banach L1(𝔾)-modules. Also, we investigate the concept of harmonic functionals over right Banach L1(𝔾)-modules and use these devices to study, among others, amenability of 𝔾.

INNER AMENABILITY OF LOCALLY COMPACT QUANTUM GROUPS

2013 ◽  
Vol 24 (07) ◽  
pp. 1350058 ◽  
Author(s):  
MOHAMMAD REZA GHANEI ◽  
RASOUL NASR-ISFAHANI
Keyword(s):  
Fixed Point ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Point Property ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Inner Amenability

We initiate a study of inner amenability for a locally compact quantum group 𝔾 in the sense of Kustermans–Vaes. We show that all amenable and co-amenable locally compact quantum groups are inner amenable. We then show that inner amenability of 𝔾 is equivalent to the existence of certain functionals associated to characters on L1(𝔾). For co-amenable locally compact quantum groups, we introduce and study strict inner amenability and its relation to the extension of the co-unit ϵ from C0(𝔾) to L∞(𝔾). We then obtain a number of equivalent statements describing strict inner amenability of 𝔾 and the existence of certain means on subspaces of L∞(𝔾) such as LUC(𝔾), RUC(𝔾) and UC(𝔾). Finally, we offer a characterization of strict inner amenability in terms of a fixed point property for L1(𝔾)-modules.

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

scholarly journals From Quantum Groups to Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1073-1094 ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthias Neufang
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Large Classes ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Recent Developments ◽  
Compact Quantum Groups ◽  
Quantum Version ◽  
Quantum Point

AbstractIn this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group 𝔾 a locally compact group 𝔾˜ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group 𝔾. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of 𝔾 are encoded by 𝔾˜; the latter, despite being a simpler object, can carry very important information about 𝔾.

scholarly journals A Note on Amenability of Locally Compact Quantum Groups

2014 ◽  
Vol 57 (2) ◽  
pp. 424-430 ◽  
Author(s):  
Piotr M. Sołtan ◽  
Ami Viselter
Keyword(s):  
Quantum Groups ◽  
Short Note ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups

AbstractIn this short note we introduce a notion called quantum injectivity of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. In particular, this provides a new characterization of amenability of locally compact groups.

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