QUANTUM GROUPS AND DUALITY

1993 ◽  
Vol 05 (02) ◽  
pp. 417-451 ◽  
Author(s):  
ELLIOT C. GOOTMAN ◽  
ALDO J. LAZAR
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Haar Measure ◽  
Duality Theory ◽  
Duality Theorem ◽  
Compact Quantum Group ◽  
Absolutely Continuous ◽  
C Algebra ◽  
One To One ◽  
The Ideal

We present an approach to the duality theory for quantum groups which is explicitly modelled on the construction of the group C*-algebra C* (G) from C0 (G), namely: in the dual of C0 (G), single out the ideal of all elements absolutely continuous with respect to Haar measure, renorm with a C*-norm determined by the representations of this ideal, and complete. We thus obtain a C*-algebra whose *-representations are in one-to-one correspondence with the representations of the quantum group. This C*-algebra is itself a (non-compact) quantum group, and we verify a duality theorem for it.

scholarly journals A REMARK ON QUANTUM GROUP ACTIONS AND NUCLEARITY

2002 ◽  
Vol 14 (07n08) ◽  
pp. 787-796 ◽  
Author(s):  
S. DOPLICHER ◽  
R. LONGO ◽  
J. E. ROBERTS ◽  
L. ZSIDÓ
Keyword(s):  
Fixed Point ◽  
Quantum Group ◽  
Haar Measure ◽  
Group Actions ◽  
Compact Quantum Group ◽  
C Algebra ◽  
Quantum Group Actions

Let H be a compact quantum group with faithful Haar measure and bounded counit. If α is an action of H on a C*- algebra A, we show that A is nuclear if and only if the fixed-point subalgebra Aα is nuclear. As a consequence H is a nuclear C*-algebra.

scholarly journals Compact quantum groups and their corepresentations

1998 ◽  
Vol 57 (1) ◽  
pp. 73-91
Author(s):  
Huu Hung Bui
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Duality Theorem ◽  
Compact Quantum Group ◽  
Compact Groups ◽  
Matrix Elements ◽  
Orthogonality Relations ◽  
The Matrix ◽  
Compact Quantum Groups ◽  
Abstract Categories

A compact quantum group is defined to be a unital Hopf C*–algebra generated by the matrix elements of a family of invertible corepresentations. We present a version of the Tannaka–Krein duality theorem for compact quantum groups in the context of abstract categories; this result encompasses the result of Woronowicz and the classical Tannaka-Krein duality theorem. We construct the orthogonality relations (similar to the case of compact groups). The Plancherel theorem is then established.

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 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 Generating Functionals for Locally Compact Quantum Groups

2020 ◽  
Author(s):  
Adam Skalski ◽  
Ami Viselter
Keyword(s):  
Quantum Group ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Generating Functional ◽  
Positive Functional ◽  
C Algebra ◽  
Locally Compact Quantum Groups ◽  
Dense Subalgebra ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

Abstract Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital *-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense *-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.

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.

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.

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