Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

2018 ◽  
Vol 29 (13) ◽  
pp. 1850092 ◽  
Author(s):  
Paweł kasprzak
Keyword(s):  
Quantum Groups ◽  
Mild Condition ◽  
Locally Compact Group ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
One To One ◽  
Scaling Group

A one-to-one correspondence between shifts of group-like projections on a locally compact quantum group [Formula: see text] which are preserved by the scaling group and contractive idempotent functionals on the dual [Formula: see text] is established. This is a generalization of the Illie–Spronk’s correspondence between contractive idempotents in the Fourier–Stieltjes algebra of a locally compact group [Formula: see text] and cosets of open subgroups of [Formula: see text]. We also establish a one-to-one correspondence between nondegenerate, integrable, [Formula: see text]-invariant ternary rings of operators [Formula: see text], preserved by the scaling group and contractive idempotent functionals on [Formula: see text]. Using our results, we characterize coideals in [Formula: see text] admitting an atom preserved by the scaling group in terms of idempotent states on [Formula: see text]. We also establish a one-to-one correspondence between integrable coideals in [Formula: see text] and group-like projections in [Formula: see text] satisfying an extra mild condition. Exploiting this correspondence, we give examples of group-like projections which are not preserved by the scaling group.

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.

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.

scholarly journals Landstad–Vaes theory for locally compact quantum groups

2018 ◽  
Vol 29 (04) ◽  
pp. 1850028
Author(s):  
Sutanu Roy ◽  
Stanisław Lech Woronowicz
Keyword(s):  
Dynamical System ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Crossed Product ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of [Formula: see text]-dynamical system introducing the concept of weak action of quantum groups on [Formula: see text]-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

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.

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 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.

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