Duality, cohomology, and geometry of locally compact quantum groups

Kalantar, Mehrdad; Neufang, Matthias

doi:10.1016/j.jmaa.2013.04.024

Cited by 7 publications

(10 citation statements)

References 18 publications

(22 reference statements)

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“…This lifting of quantum group convolution to T (L 2 (G)) allows one to study properties of G and G as well as their interactions a single space. One such interaction was obtained in [23], and states that the dual products anti-commute. For a locally compact quantum group G, the multiplications and define operator T (L 2 (G))-bimodule structures on B(L 2 (G)) such that for x ∈ L ∞ (G) and f = ω| L ∞ (G) with ω ∈ T (L 2 (G)), we have (4.5)…”

Section: And the Left And Right Fundamental Unitaries Satisfymentioning

confidence: 99%

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Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

Crann¹

2017

Can. j. math.

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Abstract. Building 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 G and -injectivity of L ∞ (Ĝ) as an operator L (Ĝ)-module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra V N(G) is -injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded L (Ĝ)-module maps on L ∞ (Ĝ), and give a simpli ed 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.

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Section: And the Left And Right Fundamental Unitaries Satisfymentioning

confidence: 99%

“…[23, Theorem 3.3] Let G be a locally compact quantum group. Then for every ρ, ω, τ ∈ T (L 2 (G)) we have(4.4) (ρ ω) τ = (ρ τ ) ω.…”

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confidence: 99%

Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

Crann¹

2017

Can. j. math.

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“…For commutative and co-commutative quantum groups, this type of multiplicative structure on T (L 2 (G)) has been studied in [1,22,23,24,26], and the general case has been investigated in [14,15,17]. In particular, it was shown in [14, Lemma 5.2] that the pre-annihilator L ∞ (G) ⊥ of L ∞ (G) in T (L 2 (G)) is a norm closed two sided ideal in (T (L 2 (G)), ⊲) and (T (L 2 (G)), ⊳), respectively, and the complete quotient map…”

Section: Luc(g) and Luc(g) *mentioning

confidence: 99%

Amenability and covariant injectivity of locally compact quantum groups

Crann

2015

Trans. Amer. Math. Soc.

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As is well known, the equivalence between amenability of a locally compact group G and injectivity of its von Neumann algebra L(G) does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if L(G) is considered as a T (L2(G))-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.2010 Mathematics Subject Classification: Primary 22D15 46L89 81R15, Secondary 43A07 46M10 43A20. 1 2 JASON CRANN 1,2 AND MATTHIAS NEUFANG 1,2 a locally compact quantum group G, as introduced in [13,29]. For any locally compact quantum group G, there are two canonical completely contractive Banach algebra structures on T (L 2 (G)), denoted by (T (L 2 (G)), ⊳) and (T (L 2 (G)), ⊲), induced by the left and right fundamental unitaries of G, respectively. This in turn yields two interesting bimodule structures on B(L 2 (G)), which have been a recent topic of interest in the development of harmonic analysis on locally compact quantum groups [14,15], and are closely related to LUC(G) and RUC(G).The dual space of LUC(G) carries a natural Banach algebra structure. In [14], Hu, Neufang and Ruan studied various properties of this algebra, in particular through a weak*-weak* continuous, injective, completely contractive representationin the algebra of completely bounded right (T (L 2 (G)), ⊲)-module maps on B(L 2 (G)). This representation is the fundamental tool in our work, and is used in section 4 to show that a locally compact quantum group G is amenable if and only if the dual quantum groupĜ is what we shall call covariantly injective, meaning the corresponding projection of norm one commutes with the module action of (T (L 2 (G)), ⊲) on B(L 2 (G)). As an application, we obtain a new proof of the recently answered question of Bédos and Tuset concerning the topological amenability of G [38]. By examining the remaining three T (L 2 (G))-module structures on B(L 2 (G)), we obtain new characterizations of amenability, co-commutativity, as well as injectivity ofĜ. Moreover, compactness of G can be characterized in terms of normal conditional expectations respecting the T (L 2 (G))-module structure.We finish in section 5 by showing that a locally compact quantum group G is amenable if and only if L ∞ (Ĝ) is an injective operator T (L 2 (G))-module. Even in the commutative case, this provides a new identification of classical amenability of a locally compact group G in terms of the injectivity of L(G) as a T (L 2 (G))-module. We also show that both amenability of G and ofĜ may be characterized through the injectivity of B(L 2 (G)) as a left, respectively, right T (L 2 (G))-module. This, along with other results in the paper suggests that these homological methods may provide a new approach to the duality problem of amenability and co-amenability for arbitrary locally compact quantum groups. PreliminariesA locally compact quantum group is a quadruple G = (L ∞ (G), Γ, ϕ, ψ), where L ∞ (G) is a ...

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“…As usual, the restriction Φ| L ∞ (G)⊗1 : L ∞ (G) → L ∞ (G) maps into C, and, moreover, it is a right L 1 (G)-module map, so G is compact by normality of Φ. Since compact quantum groups are regular, we may repeat the proof of (iii) ⇒ (i) from [14,Theorem 5.14] to deduce the discreteness of G. Thus, G is finite-dimensional by [38,Theorem 4.8].…”

Section: Proposition 52 Let G Be a Locally Compact Group Then L 1 mentioning

confidence: 95%

Inner amenability and approximation properties of locally compact quantum groups

Crann¹

2019

Indiana Univ. Math. J.

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We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau-Paterson [46], resolve a recent conjecture of Ng-Viselter [51], and prove that, for inner amenable quantum groups G, approximation properties of the dual operator algebras can be averaged to approximation properties of G. Similar homological techniques are used to prove that ℓ 1 (G) is not relatively operator biflat for any non-unimodular discrete quantum group G; a unimodular discrete quantum group G with Kirchberg's factorization property is weakly amenable if and only if L 1 cb ( G) is operator amenable, and amenability of a locally compact quantum group G implies Cu( G) ∼ = L 1 ( G) ⊗ L 1 ( G) C0( G) completely isometrically. The latter result allows us to partially answer a conjecture of Voiculescu [67] when G has the approximation property.2010 Mathematics Subject Classification. Primary 22D35; Secondary 46M10, 46L89.

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Duality, cohomology, and geometry of locally compact quantum groups

Cited by 7 publications

References 18 publications

Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

Amenability and covariant injectivity of locally compact quantum groups

Inner amenability and approximation properties of locally compact quantum groups

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