Young's inequality for locally compact quantum groups

Journal of Mathematical Physics

2017

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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.

Section: Resultsmentioning

confidence: 98%

“…For any x in L 1 (G), we deonte by xϕ the bounded linear functional on L ∞ (G) given by (xϕ)(y) = ϕ(yx) for any y in L ∞ (G). Recall that a projection p in [20] for more properties of biprojections).…”

Section: Preliminariesmentioning

confidence: 99%

Uncertainty principles for Kac algebras

Journal of Mathematical Physics

2017

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“…The quantum inequalities in Theorem 2.1 on these infinite quantum symmetries have been partially studied in refs. [36][37][38][39][40]. The quantum uncertainty principle QUP -2 in Theorem 2.1 becomes a continuous family of inequalities on locally compact quantum groups (40).…”

Section: Qfa On Locally Compact Quantum Groupsmentioning

confidence: 99%

Quantum Fourier analysis

Jaffe

Jiang

Proc. Natl. Acad. Sci. U.S.A.

et al. 2020

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Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined Lp spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

“…For the infinite dimensional case, Kusterman and Vaes introduced locally compact quantum groups [16]. Young's inequality for locally compact quantum groups was proved in [21]. It would be interesting to characterize the extremal pairs.…”

Section: Extremal Pairs Of Young's Inequalitymentioning

confidence: 99%

Block maps and Fourier analysis

Jiang

2019

Sci. China Math.

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We introduce block maps for subfactors and study their dynamic systems. We prove that the limit points of the dynamic system are positive multiples of biprojections and zero. For the Z 2 case, the asymptotic phenomenon of the block map coincides with that of that 2D Ising model. The study of block maps requires a further development of the recent work of the authors on the Fourier analysis of subfactors. We generalize the notion of sum set estimates in additive combinatorics for subfactors and prove the exact inverse sum set theorem. Using this new method, we characterize the extremal pairs of Young's inequality for subfactors, as well as the extremal operators of the Hausdorff-Young inequality.