Amenability of compact hypergroup algebras

Amini, Massoud; Medghalchi, Alireza

doi:10.1002/mana.201200284

Cited by 7 publications

(5 citation statements)

References 11 publications

(20 reference statements)

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“…where f = 0 is a function in L 2 (K, µ K ) and f is its Fourier transform, µ K is the Haar measure on the compact group or DJS hypergroup, n ρ is the dimension of the irreducible representation ρ and k ρ is its hyperdimension [Vre79], [AM14]. Note that for compact groups n ρ = k ρ , and for subfactor theoretical compact hypergroups n ρ ≤ k ρ = d(ρ) [BDVG21, Cor.…”

Section: Inversion Formula and Uncertainty Principlesmentioning

confidence: 99%

Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

Bischoff,

Del Vecchio,

Giorgetti

2021

Preprint

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Discrete subfactors include a particular class of infinite index subfactors and all finite index ones. A discrete subfactor is called local when it is braided and it fulfills a commutativity condition motivated by the study of inclusion of Quantum Field Theories in the algebraic Haag-Kastler setting. In [BDVG21], we proved that every irreducible local discrete subfactor arises as the fixed point subfactor under the action of a canonical compact hypergroup. In this work, we prove a Galois correspondence between intermediate von Neumann algebras and closed subhypergroups, and we study the subfactor theoretical Fourier transform in this context. Along the way, we extend the main results concerning α-induction and σ-restriction for braided subfactors previously known in the finite index case. Contents1. Introduction 2. Preliminaries 2.1. The canonical endomorphism 2.2. Conditional expectations 2.3. Braided and local subfactors 2.4. Compact hypergroups 2.5. Duality theorem and dominated UCP maps 3. α-induction for discrete subfactors 4. Intermediate inclusions 5. Galois correspondence 6. Fourier transform 6.1. Extension of the Fourier transform to UCP maps 6.2. The local discrete case: Fourier transform on measures 6.3. L p spaces and Fourier inequalities 6.4. Involutions, convolutions and products 6.5. Convolution inequalities 6.6. Inversion formula and uncertainty principles References

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Section: Inversion Formula and Uncertainty Principlesmentioning

confidence: 99%

Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

Bischoff,

Del Vecchio,

Giorgetti

2021

Preprint

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“…Here we briefly mention the gaps in the argument presented in [6] using its notation and context. In the first computation line on p. 1617, we should have…”

Section: Application To Compact Groupsmentioning

confidence: 99%

“…Remark 4.7 The main theorem of [6], which is stated for compact hypergroups, claims to prove both sides of [7, Conjecture 0.1], and, in particular, Corollary 4.5, using the theory of compact hypergroups. Unfortunately, there are gaps in each direction of the proof of that result.…”

Section: Application To Compact Groupsmentioning

confidence: 99%

Fourier algebras of hypergroups and central algebras on compact (quantum) groups

Alaghmandan¹,

Crann²

2017

Studia Math.

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This paper concerns the study of regular Fourier hypergroups through multipliers of their associated Fourier algebras. We establish hypergroup analogues of well-known characterizations of group amenability, introduce a notion of weak amenability for hypergroups, and show that every discrete commutative hypergroup is weakly amenable with constant 1. Using similar techniques, we provide a sufficient condition for amenability of hypergroup Fourier algebras, which, as an immediate application, answers one direction of a conjecture of Azimifard-Samei-Spronk [7] on the amenability of ZL 1 (G) for compact groups G. In the final section we consider Fourier algebras of hypergroups arising from compact quantum groups G, and in particular, establish a completely isometric isomorphism with the center of the quantum group algebra for compact G of Kac type.

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“…In particular, we show that Trig(N ⊂ M) can be identified with the algebra of trigonometric polynomials in the sense of Vrem [Vre79]. Moreover, we prove the equality between the hyperdimension of a representation [AM14] and the dimension of the associated endomorphism ρ ≺ θ [LR97] (Theorem 6.5).…”

Section: Introductionmentioning

confidence: 99%

Compact Hypergroups from Discrete Subfactors

Bischoff,

Del Vecchio,

Giorgetti

2020

Preprint

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Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra.We show that to every irreducible local discrete subfactor N ⊂ M of type III there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on M by unital completely positive (ucp) maps and which gives N as fixed points. To show this, we establish a duality pairing between the set of all N -bimodular ucp maps on M and a certain commutative unital C * -algebra, whose spectrum we identify with the compact hypergroup.If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact quantum groups acting as global gauge symmetries in local conformal field theory. 6. Representation theory 7. The case of depth 2: No compact quantum group orbifolds 8. A remark on non-local extensions 9. Examples 9.1. Group orbifolds 9.2. Double coset orbifolds 9.3. Quantum channels with infinite index 9.4. Local discrete subfactors in Conformal Field Theory 9.5. Constructions of discrete subfactors References

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Amenability of compact hypergroup algebras

Cited by 7 publications

References 11 publications

Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

Fourier algebras of hypergroups and central algebras on compact (quantum) groups

Compact Hypergroups from Discrete Subfactors

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