scholarly journals A MEAN ERGODIC THEOREM FOR ACTIONS OF AMENABLE QUANTUM GROUPS

2008 ◽  
Vol 78 (1) ◽  
pp. 87-95 ◽  
Author(s):  
ROCCO DUVENHAGE
Keyword(s):  
Ergodic Theorem ◽  
Quantum Groups ◽  
Von Neumann Algebra ◽  
Weak Form ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Mean Ergodic Theorem ◽  
Compact Quantum Groups ◽  
The Mean

AbstractWe prove a weak form of the mean ergodic theorem for actions of amenable locally compact quantum groups in the von Neumann algebra setting.

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 Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

2017 ◽  
Vol 69 (5) ◽  
pp. 1064-1086 ◽  
Author(s):  
Jason Crann
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Relative Homology ◽  
Convolution Algebras ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Group Von Neumann Algebra

AbstractBuilding on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group and 1-injectivity of as an operator -module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded -module maps on , and give a simpliûed proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

scholarly journals REPRESENTATION OF LEFT CENTRALIZERS FOR ACTIONS OF LOCALLY COMPACT QUANTUM GROUPS

2013 ◽  
Vol 24 (04) ◽  
pp. 1350025 ◽  
Author(s):  
MEHRDAD KALANTAR
Keyword(s):  
Quantum Groups ◽  
Representation Theorem ◽  
Von Neumann Algebras ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups

We generalize the representation theorem of Junge, Neufang and Ruan in [A representation theorem for locally compact quantum groups, Internat. J. Math.20(3) (2009) 377–400], and some of the important results which were used in its proof, to the case of actions of locally compact quantum groups on von Neumann algebras.

scholarly journals Locally compact quantum groups in the von Neumann algebraic setting

2003 ◽  
Vol 92 (1) ◽  
pp. 68 ◽  
Author(s):  
Johan Kustermans ◽  
Stefaan Vaes
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Algebraic Setting ◽  
Locally Compact Quantum Groups ◽  
Algebraic Quantum ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a $C^*$-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper [8]. We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its dual. Loosely speaking we show how the antipode of the locally compact quantum group determines the modular group and modular conjugation of the dual locally compact quantum group.

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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