A Hausdorff–young Inequality for Locally Compact Quantum Groups

Cooney, Tom

doi:10.1142/s0129167x10006677

Cited by 17 publications

(22 citation statements)

References 10 publications

(13 reference statements)

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“…One could define the convolution with respect to the right Haar weight when the noncommutative L p space is taken with respect to the right Haar weight. We also give the definition of shifts of group-like projections and show that they are extremal element for the Hausdorff-Young inequality given in [4]. Similar results for Young's inequality is also obtained.…”

Section: Main 12 (Theorem 313)supporting

confidence: 64%

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Young's inequality for locally compact quantum groups

Liu¹,

Wang²,

Wu³

2017

J. Operator Theory

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Section: Main 12 (Theorem 313)supporting

confidence: 64%

“…Suppose h is a group-like projection in L ϕ . Since h is analytic with respect to σ ϕ and λ(hϕ) is analytic with respect toσφ, we have that hd [4]. Now…”

Section: By Proposition 24 In [15] We Have That (Hϕ)mentioning

confidence: 91%

Young's inequality for locally compact quantum groups

Liu¹,

Wang²,

Wu³

2017

J. Operator Theory

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“…By Lemma 18 in [26], we have that α → |x| α is differentiable for α > 0. Now differentiating the Hausdorff-Young inequality [6] F…”

Section: Resultsmentioning

confidence: 99%

“…the trace of the left hand side of inequality (3) is w 6 2 . By Equation (4), we have the trace of the right hand side of inequality (3) is w 6 2 . This implies that (w * R(w) * )(w * * R(w)) = w 2 2 (ww * ) * (R(w) * R(w)).…”

Section: Now By Hölder's Inequality and Hausdorff-young Inequality [6]mentioning

confidence: 99%

Uncertainty principles for Kac algebras

Liu

2017

Journal of Mathematical Physics

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In this paper, we introduce the notation of bi-shift of biprojections in subfactor theory to unimodular Kac algebras. We characterize the minimizers of Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle for unimodular Kac algebras with biprojections and prove Hardy's uncertainty principle in terms of minimizers. 1 Introduction Uncertainty principles for locally compact abelian groups were studied by Hardy [15], Hirschman [16], Beckner [2], Donoho and Stark [9], Smith [23], Tao [24] etc. In 2008, Alagic and Russell [1] proved Donoho-Stark uncertainty principle for compact groups. In 2004,Özaydm and Przebinda [21] characterized the minimizers of Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle for locally compact abelian groups.Kac algebras were introduced independently by L.I Vainerman and G.I. Kac [27,28,29] and by Enock and Nest [10,11,12], which generalized locally compact groups and their duals. Furthermore, J. Kustermans and S. Vaes introduced locally compact quantum groups [18]. Recently Crann and Kalantar proved Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle for unimodular locally compact quantum groups [7].Subfactor theory also provides a natural framework to study quantum symmetry. The group symmetry is captured by the subfactor arisen from the group crossed product construction. Ocneanu first pointed out the one-to-one correspondence between finite dimensional Kac algebras and finiteindex, depth-two, irreducible subfactors. This correspondence was proved by W. Szymanski [22]. Enock and Nest generalized the correspondence to infinite dimensional compact (or discrete) type Kac algebras and infinite-index, depth-two, irreducible subfactors [14]. In general, a subfactor provides a pair of non-commutative spaces dual to each other and a Fourier transform F between them. It appears to be natural to study Fourier analysis for subfactors.

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“…The Fourier transform for locally compact quantum groups has been discussed in [Coo10], [Cas13] and [Kah10]. In the setting of compact quantum groups, we may give a more explicit description.…”

Section: Fourier Series and Multipliersmentioning

confidence: 99%

Lacunary Fourier Series for Compact Quantum Groups

Wang

2016

Commun. Math. Phys.

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Abstract. This paper is devoted to the study of Sidon sets, Λ(p)-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, Λ(p)-sets and lacunarities for L pFourier multipliers, generalizing a previous work by Blendek and Michalicek. We also prove the existence of Λ(p)-sets for orthogonal systems in noncommutative L p -spaces, and deduce the corresponding properties for compact quantum groups. Central Sidon sets are also discussed, and it turns out that the compact quantum groups with the same fusion rules and the same dimension functions have identical central Sidon sets. Several examples are also included.

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A Hausdorff–young Inequality for Locally Compact Quantum Groups

Cited by 17 publications

References 10 publications

Young's inequality for locally compact quantum groups

Young's inequality for locally compact quantum groups

Uncertainty principles for Kac algebras

Lacunary Fourier Series for Compact Quantum Groups

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