Inner amenability and approximation properties of locally compact quantum groups

Crann, Jason

doi:10.1512/iumj.2019.68.7829

Cited by 10 publications

(7 citation statements)

References 63 publications

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“…Remark 5.3. In [8], Crann defined a notion of inner amenability for quantum groups as the existence of an invariant state for the canonical action…”

Section: Theorem 52 Let G Be a Discrete Kac Algebra Then G Is Amenabl...mentioning

confidence: 99%

Amenable actions of discrete quantum groups on von Neumann algebras

Moakhar

2018

Preprint

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We introduce the notion of Zimmer amenability for actions of discrete quantum groups on von Neumann algebras. We prove generalizations of several fundamental results of the theory in the noncommutative case. In particular, we give a characterization of Zimmer amenability of an action α : G N in terms of Ĝ-injectivity of the von Neumann algebra crossed product N ⋉α G. As an application we show that the actions of any discrete quantum group on its Poisson boundaries are always amenable.Contents 20 References 24

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“…Remark 5.3. In [8], Crann defined a notion of inner amenability for quantum groups as the existence of an invariant state for the canonical action…”

Section: Theorem 52 Let G Be a Discrete Kac Algebra Then G Is Amenabl...mentioning

confidence: 99%

Amenable actions of discrete quantum groups on von Neumann algebras

Moakhar

2018

Preprint

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“…The following tool will be used heavily in the sequel when computing specific examples. The proof technique is similar to [8,Corollary 7.4]. Proposition 3.3.…”

Section: Haagerup Duality For Operator Modulesmentioning

confidence: 99%

A new duality via the Haagerup tensor product

Alaghmandan

Crann

Neufang

2019

Journal of Mathematical Analysis and Applications

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We initiate the study of a new notion of duality defined with respect to the module Haagerup tensor product. This notion not only recovers the standard operator space dual for Hilbert C * -modules, it also captures quantum group duality in a fundamental way. We compute the socalled Haagerup dual for various operator algebras arising from ℓ p spaces. In particular, we show that the dual of ℓ 1 under any operator space structure is min ℓ ∞ . In the setting of abstract harmonic analysis we generalize a result of Varopolous by showing that C(G) is an operator algebra under convolution for any compact Kac algebra G. We then prove that the corresponding Haagerup dual C(G) h = ℓ ∞ ( G), whenever G is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.

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“…When G is co-commutative, the equivalence of (1) and (3) is simply an application of Leptin's theorem [32]. When G has the approximation property, the result follows from [14,Corollary 7.4].…”

Section: Proof Pick Diagramsmentioning

confidence: 99%

“…In the setting of operator modules, the author's recent work [12][13][14][15] characterizes important properties of a locally compact quantum group G in terms of homological properties of various operator modules over the convolution algebra L 1 (G) (or its dual). For instance, G is amenable if and only if the dual von Neumann algebra L ∞ ( G) is injective over L 1 ( G) [12,Theorem 5.1].…”

Section: Introductionmentioning

confidence: 99%

Finite presentation, the local lifting property, and local approximation properties of operator modules

Crann¹

2020

Preprint

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We introduce notions of finite presentation and co-exactness which serve as qualitative and quantitative analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we begin the development of a local theory of operator modules by introducing analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules we prove that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan [30]. We pursue applications to abstract harmonic analysis, where, for a locally compact quantum group G, we show that L 1 (G)-nuclearity of LUC(G) and L 1 (G)-semi-discreteness of L ∞ (G) are both equivalent to co-amenability of G. We establish the equivalence between A(G)-injectivity of G ⋉M , A(G)-semi-discreteness of G ⋉M , and amenability of W * -dynamical systems (M, G, α) with M injective. We end with remarks on future directions.

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Inner amenability and approximation properties of locally compact quantum groups

Cited by 10 publications

References 63 publications

Amenable actions of discrete quantum groups on von Neumann algebras

Amenable actions of discrete quantum groups on von Neumann algebras

A new duality via the Haagerup tensor product

Finite presentation, the local lifting property, and local approximation properties of operator modules

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