scholarly journals POISSON BOUNDARIES OVER LOCALLY COMPACT QUANTUM GROUPS

2013 ◽  
Vol 24 (03) ◽  
pp. 1350023 ◽  
Author(s):  
MEHRDAD KALANTAR ◽  
MATTHIAS NEUFANG ◽  
ZHONG-JIN RUAN
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Harmonic Functions ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Concrete Realization ◽  
Separable Case ◽  
Noncommutative Analogue

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU q(2) arising from measures on its spectrum.

scholarly journals Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups

2016 ◽  
Vol 68 (2) ◽  
pp. 309-333 ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Multiplier Algebra ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Group Form ◽  
Classical Quantum

AbstractWe show that the assignment of the (left) completely bounded multiplier algebra Mlcb(L1()) to a locally compact quantum group , and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf *-homomorphisms between universal C*-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal C*-algebra level, and that the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms then interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal C*-algebra picture, and then, again, how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the “maximal classical” quantum subgroup of a locally compact quantum group, that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Amenability and Co-Amenability for Locally Compact Quantum Groups

2003 ◽  
Vol 14 (08) ◽  
pp. 865-884 ◽  
Author(s):  
E. Bédos ◽  
L. Tuset
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Weak Containment ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqg's.

scholarly journals Compact Operators in Regular LCQ Groups

2014 ◽  
Vol 57 (3) ◽  
pp. 546-550 ◽  
Author(s):  
Mehrdad Kalantar
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Operators ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Nikodym Property ◽  
Compact Quantum Groups

AbstractWe show that a regular locally compact quantum group 𝔾 is discrete if and only if 𝓛∞(𝔾) contains non-zero compact operators on 𝓛2(𝔾). As a corollary we classify all discrete quantum groups among regular locally compact quantum groups 𝔾 where 𝓛1(𝔾) has the Radon-Nikodym property.

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.

A Homological Property and Arens Regularity of Locally Compact Quantum Groups

2017 ◽  
Vol 60 (1) ◽  
pp. 122-130 ◽  
Author(s):  
Mohammad Reza Ghanei ◽  
Rasoul Nasr-Isfahani ◽  
Mehdi Nemati
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Group Algebra ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Homological Property ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Arens Regularity

AbstractWe characterize two important notions of amenability and compactness of a locally compact quantum group G in terms of certain homological properties. For this, we show that G is character amenable if and only if it is both amenable and co-amenable. We ûnally apply our results to Arens regularity problems of the quantum group algebra L1(G). In particular, we improve an interesting result by Hu, Neufang, and Ruan.

scholarly journals EVERY TOPOLOGICALLY AMENABLE LOCALLY COMPACT QUANTUM GROUP IS AMENABLE

2012 ◽  
Vol 87 (1) ◽  
pp. 149-151 ◽  
Author(s):  
AMIN ZOBEIDI
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Open Problem ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

AbstractWe prove that every topologically amenable locally compact quantum group is amenable. This answers an open problem by Bédos and Tuset [‘Amenability and co-amenability for locally compact quantum groups’, Internat. J. Math.14 (2003), 865–884].

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 DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS

2005 ◽  
Vol 4 (1) ◽  
pp. 135-173 ◽  
Author(s):  
Saad Baaj ◽  
Stefaan Vaes
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Matched Pair ◽  
Crossed Products ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Algebraic Properties ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

For a matched pair of locally compact quantum groups, we construct the double crossed product as a locally compact quantum group. This construction generalizes Drinfeld’s quantum double construction. We study the modular theory and the $\mathrm{C}^*$-algebraic properties of these double crossed products, as well as several links between double crossed products and bicrossed products. In an appendix, we study the Radon–Nikodym derivative of a weight under a quantum group action (following Yamanouchi) and obtain, as a corollary, a new characterization of closed quantum subgroups. AMS 2000 Mathematics subject classification: Primary 46L89. Secondary 46L65

scholarly journals LOCALLY COMPACT QUANTUM GROUPS IN THE UNIVERSAL SETTING

2001 ◽  
Vol 12 (03) ◽  
pp. 289-338 ◽  
Author(s):  
JOHAN KUSTERMANS
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Group Structure ◽  
Locally Compact ◽  
Bijective Correspondence ◽  
Locally Compact Quantum Groups ◽  
Algebraic Quantum ◽  
Compact Quantum Groups ◽  
Group A ◽  
Fine Tune

In this paper we associate to every reduced C *-algebraic quantum group (A, Δ) (as defined in [11]) a universal C *-algebraic quantum group (Au, Δu). We fine tune a proof of Kirchberg to show that every *-representation of a modified L 1-space is generated by a unitary corepresentation. By taking the universal enveloping C *-algebra of a dense sub *-algebra of A we arrive at the C *-algebra Au. We show that this C *-algebra Au carries a quantum group structure which is a rich as its reduced companion. We also establish a bijective correspondence between quantum group morphisms and certain co-actions.

Property T for locally compact quantum groups

2015 ◽  
Vol 26 (03) ◽  
pp. 1550024 ◽  
Author(s):  
Xiao Chen ◽  
Chi-Keung Ng
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Short Paper ◽  
Locally Compact ◽  
C Algebras ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Dimensional Irreducible Representation ◽  
Property T

In this short paper, we obtained some equivalent formulations of property T for a general locally compact quantum group 𝔾, in terms of the full quantum group C*-algebras [Formula: see text] and the *-representation of [Formula: see text] associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if 𝔾 is of Kac type, we show that 𝔾 has property T if and only if every finite-dimensional irreducible *-representation of [Formula: see text] is an isolated point in the spectrum of [Formula: see text] (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property T discrete quantum groups using bicrossed products.

Sign in / Sign up

Export Citation Format

Share Document