On Weak Containment Properties

Rindler, Harald

doi:10.2307/2159681

Cited by 5 publications

(5 citation statements)

References 4 publications

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“…Then G is called a tall group . Some properties of tall groups, specially profinite tall groups, have been studied in . Example Let us use the notations and facts mentioned in the proof of Proposition .…”

Section: Amenability Of Hypergroup Algebrasmentioning

confidence: 99%

Amenability notions of hypergroups and some applications to locally compact groups

Alaghmandan

2017

Math. Nachr.

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Different notions of amenability on hypergroups and their relations are studied. Developing Leptin's theorem for discrete hypergroups, we characterize the existence of a bounded approximate identity for hypergroup Fourier algebras. We study the Leptin condition for discrete hypergroups derived from the representation theory of some classes of compact groups. Studying amenability of the hypergroup algebras for discrete commutative hypergroups, we obtain some results on amenability properties of some central Banach algebras on compact and discrete groups.Keyword: hypergroups; Fourier algebra; amenability; compact groups; finite conjugacy groups.AMS codes: 43A62, 46H20.In this paper, we investigate different amenability notions of hypergroups and their relations. In particular, we look closer at a generalization of Følner type conditions over hypergroups. This generalization lets us to investigate more structural properties of some classes of hypergroups. In particular we study the existence of bounded approximate identities for the Fourier algebra of regular Fourier hypergroups. We prove a hypergroup analogue of Leptin's theorem for discrete regular Fourier hypergroups that is, the existence of a bounded approximate identity for the Fourier algebra is equivalent to the existence of a square integrable Reiter's net. The later condition is denoted by (P 2 ).We also study amenability of hypergroup algebras (as Banach algebras) for discrete commutative hypergroups satisfying (P 2 ). We show that for this class of hypergroups, the hypergroup algebra cannot be amenable if the inverse of the Haar measure vanishes at infinity. This result and the appearance of (P 2 ) in hypergroup Leptin's theorem emphasize the importance of the amenability notion (P 2 ). At the end we apply these results to some examples of hypergroup structures which are admitted by two classes of locally compact groups, namely compact and discrete groups.1

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Section: Amenability Of Hypergroup Algebrasmentioning

confidence: 99%

Amenability notions of hypergroups and some applications to locally compact groups

Alaghmandan

2017

Math. Nachr.

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“…where CK L 1 (G) (L 1 (G), L ∞ (G)) are the completely compact convolution maps, and then argue via (3). The isomorphisms (3) and ( 4) complement the analysis of [12,22,57], where, contrary to the classical setting, it was shown that, in general, complete compactness was insufficient to recover the quantum Bohr compactification. Our results imply that for a large class of quantum groups one can recover the (discrete dual of the) quantum Bohr compactification by considering completely nuclear convolution maps, as opposed to completely compact ones.…”

Section: Introductionmentioning

confidence: 88%

“…That is, one cannot recover the quantum Bohr compactification by considering (completely) compact module maps L 1 (G) → L ∞ (G). Indeed, by [22, §4.1],[12, Proposition 3.1] and [57, Proposition 1], if G is any infinite tall compact group with the mean-zero weak containment property (see [12,57]) then the rank-one projection p ∈ V N (G) onto the constant functions in L 2 (G) defines a completely compact A(G)-module map, but…”

Section: Completely Nuclear Multipliers and The Quantum Bohr Compacti...mentioning

confidence: 99%

Mapping ideals of quantum group multipliers

Alaghmandan¹,

Crann²,

Neufang³

2018

Preprint

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We study the dual relationship between quantum group convolution maps L 1 (G) → L ∞ (G) and completely bounded multipliers of G. For a large class of locally compact quantum groups G we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with M cb (L 1 ( G)), yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coinciding with ℓ 1 ( bG), where bG is the quantum Bohr compactification of G. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with C(bG). Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the noncommutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of quasi-SIN quantum group, which we introduce and exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras V N (G) for quasi-SIN locally compact groups G.

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“…. This is [9,Theorem 3.5], but be aware of some errors in preliminary results; these errors are partly corrected in [40]; in particular [40, Proposition 1] shows the result we are interested in. Remarkably, in full generality, it is still unknown if AP ( Ĝ) is even a C * -algebra, never-mind whether it satisfies any obvious interpretation as a "compactification".…”

Section: There Exists a Compact Groupmentioning

confidence: 99%

Remarks on the quantum Bohr compactification

Daws¹

2013

Illinois J. Math.

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The category of locally compact quantum groups can be described as either Hopf * -homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how So ltan's quantum Bohr compactification can be used to construct a "compactification" in this category. Depending on the viewpoint, different C * -algebraic compact quantum groups are produced, but the underlying Hopf * -algebras are always, canonically, the same. We show that a complicated range of behaviours, with C * -completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of So ltan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in L ∞ (G), involving the antipode, which allows one to compute the Hopf * -algebra of the compactification of G; we later study when the antipode assumption can be dropped. In the cocommutative case we make a detailed study of Runde's notion of a completely almost periodic functional-with a slight strengthening, we show that for [SIN] groups this does recover the Bohr compactification ofĜ.

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On Weak Containment Properties

Cited by 5 publications

References 4 publications

Amenability notions of hypergroups and some applications to locally compact groups

Amenability notions of hypergroups and some applications to locally compact groups

Mapping ideals of quantum group multipliers

Remarks on the quantum Bohr compactification

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