Amenable actions of discrete quantum groups on von Neumann algebras

Moakhar, Mohammad S. M.

doi:10.48550/arxiv.1803.04828

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…This will appear in future work. A similar result for actions of discrete quantum groups on von Neumann algebras was obtained independently in [35].…”

Section: Amenable Actions and The A(g)-wepsupporting

confidence: 73%

A weak expectation property for operator modules, injectivity and amenable actions

Bearden¹,

Crann²

2020

Preprint

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We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras A. We prove a number of general results-for example, a characterization of the A-WEP in terms of an appropriate A-injective envelope, and also a characterization of those A for which A-WEP implies WEP. In the case of A = L 1 (G), we recover the G-WEP for G-C * -algebras in recent work of Buss-Echterhoff-Willett [8]. When A = A(G), we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related in the setting of dynamical systems, where we show that a W * -dynamical system (M, G, α) with M injective is amenable if and only if M is L 1 (G)-injective if and only if the crossed product G ⋉M is A(G)-injective. Analogously, we show that a C * -dynamical system (A, G, α) with A nuclear and G exact is amenable if and only if A has the L 1 (G)-WEP if and only if the reduced crossed product G ⋉ A has the A(G)-WEP.

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“…This will appear in future work. A similar result for actions of discrete quantum groups on von Neumann algebras was obtained independently in [35].…”

Section: Amenable Actions and The A(g)-wepsupporting

confidence: 73%

A weak expectation property for operator modules, injectivity and amenable actions

Bearden¹,

Crann²

2020

Preprint

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“…Our results also shed insight into potential notions of (topologically) amenable co-actions of arbitrary locally compact quantum groups, building on the existing notions in the discrete case [14,23].…”

Section: Introductionmentioning

confidence: 76%

Amenable dynamical systems over locally compact groups

Bearden¹,

Crann²

2020

Preprint

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We establish a Reiter property for amenable W * -dynamical systems (M, G, α) over arbitrary locally compact groups. We also prove that a commutative C * -dynamical system (C 0 (X), G, α) is topologically amenable if and only if its universal W * -dynamical system is amenable. Our results answer three open questions from the literature; one of Anantharaman-Delaroche from [3], and two from a recent preprint of Buss-Echterhoff-Willett [9].

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“…A discrete quantum group is amenable, if and only if there exists an invariant state under an amenable CFA action. This theorem, which provides a positive answer to the question Q2 above, can be considered as a C * -algebraic analogue of [Moa18,Theorem 4.7], and in fact is also another quantum version of [Anant93,Propostion 3.6].…”

Section: Introductionmentioning

confidence: 98%

“…Viewing a von Neumann algebra as a quantum measure space, the author in [Anant93,Proposition 3.6] answered Q1 in the von Neumman algebraic setting, that is, a locally compact group G is amenble if and only if there is an G-invariant mean under an amenable action of G on a von Neumann algebra. Recently, in [Moa18], the above result [Anant93,Proposition 3.6] had been generalized to the discrete quantum group case. Since a C * -algebra is thought of as a quantum topological space, it is then a natural question to propose a C * -algebraic and quantum analogue of the question Q1 as follows.…”

Section: Introductionmentioning

confidence: 99%

Amenable fusion algebraic actions of discrete quantum groups on compact quantum spaces

Chen¹,

Goswami²,

Huang³

2021

Preprint

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In this paper, we introduce actions of fusion algebras on unital C *algebras. Considering the co-representation ring of a compact quantum group as a fusion algebra, we define, as a special class of fusion algebraic actions, a newtype "point by point" action, called a canonical fusion algebraic (for short, CFA) action, of a discrete quantum group on a compact quantum space. Also we show that this CFA action and the commonly used action can be induced by each other. Furthermore, we define amenable CFA actions, and present some basic connections between amenable CFA actions and amenable discrete quantum groups. As an application, thinking of a state on a unital C * -algebra as a "probability measure" on a compact quantum space, we show that amenability for a discrete quantum group is equivalent to both of amenability for a CFA action and the existence of this kind of "probability measure" that is invariant under this action.

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Amenable actions of discrete quantum groups on von Neumann algebras

Cited by 3 publications

References 18 publications

A weak expectation property for operator modules, injectivity and amenable actions

A weak expectation property for operator modules, injectivity and amenable actions

Amenable dynamical systems over locally compact groups

Amenable fusion algebraic actions of discrete quantum groups on compact quantum spaces

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