Locally compact quantum groups

Caspers, Martijn

doi:10.4064/bc111-0-5

Cited by 3 publications

(2 citation statements)

References 24 publications

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“…In this subsection we recall the definition of locally compact quantum groups in the sense of Kustermans and Vaes [KV00,KV03]. See also the notes [Cas17]. We shall mainly work with the von Neumann algebraic version.…”

Section: Preliminariesmentioning

confidence: 99%

$L_p$-$L_q$ Fourier multipliers on locally compact quantum groups

Zhang¹

2022

Preprint

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Let G be a locally compact quantum group with dual p G. Suppose that the left Haar weight ϕ and the dual left Haar weight p ϕ are tracial, e.g. G is a unimodular Kac algebra. We prove that for 1 ă p ď 2 ď q ă 8, the Fourier multiplier mx is bounded from Lpp p G, p ϕq to Lqp p G, p ϕq whenever the symbol x lies in Lr,8pG, ϕq, where 1{r " 1{p ´1{q. Moreover, we havewhere cp,q is a constant depending only on p and q. This was first proved by Hörmander [Hör60] for R n , and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of Lp-Fourier multipliers over discrete group von Neumann algebras. A similar result for Sp-Sq Schur multipliers is also proved.

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Section: Preliminariesmentioning

confidence: 99%

$L_p$-$L_q$ Fourier multipliers on locally compact quantum groups

Zhang¹

2022

Preprint

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“…Let us first recall some well-known definitions and properties concerning compact quantum groups. We refer to [Cas1], [FSS1], [Kus1], [MaD1], [NeT1], [VDa1] and references therein for more details.…”

Section: Compact Quantum Groupsmentioning

confidence: 99%

Quantum information theory and Fourier multipliers on quantum groups

Arhancet¹

2020

Preprint

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In this paper, we compute the exact value of the minimum output entropy of all quantum channels induced by Fourier multipliers acting on an arbitrary finite quantum group G. Moreover, we show that this quantity is equal to the (normalized) completely bounded minimal entropy. We also give a value for the classical capacity and the quantum capacity. Our results rely on a new and precise description of bounded Fourier multipliers from L 1 (G) into L p (G) for 1 < p ∞ where G is a co-amenable compact quantum group of Kac type and on the automatic completely boundedness of these multipliers that this description entails. This result is new even in the case of a group von Neumann algebra associated to an amenable discrete group. Contents 1 Introduction 1 2 Preliminaries 4 3 Convolutions by elements of L p (G) and c.b. maps from L 1 (G) into L p (G) 8 4 Bounded versus complete. bounded Fourier multipliers from L 1 (G) into L p (G) 10 5 Bounded versus completely bounded operators from L 1 (M) into L p (M) 16 6 ℓ p (S p d )-summing Fourier multipliers on L ∞ (G) 20 7 Entropies and capacities 22 8 Appendix: output minimal entropy as derivative of operator norms 23 Bibliography 25

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$$L_p$$–$$L_q$$ Fourier Multipliers on Locally Compact Quantum Groups

Zhang

2023

J Fourier Anal Appl

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Let $$\mathbb {G}$$ G be a locally compact quantum group with dual $$\widehat{\mathbb {G}}$$ G ^ . Suppose that the left Haar weight $$\varphi $$ φ and the dual left Haar weight $$\widehat{\varphi }$$ φ ^ are tracial, e.g. $$\mathbb {G}$$ G is a unimodular Kac algebra. We prove that for $$1<p\le 2 \le q<\infty $$ 1 < p ≤ 2 ≤ q < ∞ , the Fourier multiplier $$m_{x}$$ m x is bounded from $$L_p(\widehat{\mathbb {G}},\widehat{\varphi })$$ L p ( G ^ , φ ^ ) to $$L_q(\widehat{\mathbb {G}},\widehat{\varphi })$$ L q ( G ^ , φ ^ ) whenever the symbol x lies in $$L_{r,\infty }(\mathbb {G},\varphi )$$ L r , ∞ ( G , φ ) , where $$1/r=1/p-1/q$$ 1 / r = 1 / p - 1 / q . Moreover, we have $$\begin{aligned} \Vert m_{x}:L_p(\widehat{\mathbb {G}},\widehat{\varphi })\rightarrow L_q(\widehat{\mathbb {G}},\widehat{\varphi })\Vert \le c_{p,q} \Vert x\Vert _{L_{r,\infty }(\mathbb {G},\varphi )}, \end{aligned}$$ ‖ m x : L p ( G ^ , φ ^ ) → L q ( G ^ , φ ^ ) ‖ ≤ c p , q ‖ x ‖ L r , ∞ ( G , φ ) , where $$c_{p,q}$$ c p , q is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93–140, 1960) for $$\mathbb {R}^n$$ R n , and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of $$L_p$$ L p -Fourier multipliers over discrete group von Neumann algebras. A similar result for $$\mathcal {S}_p$$ S p -$$\mathcal {S}_q$$ S q Schur multipliers is also proved.

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Locally compact quantum groups

Cited by 3 publications

References 24 publications

$L_p$-$L_q$ Fourier multipliers on locally compact quantum groups

$L_p$-$L_q$ Fourier multipliers on locally compact quantum groups

Quantum information theory and Fourier multipliers on quantum groups

$$L_p$$–$$L_q$$ Fourier Multipliers on Locally Compact Quantum Groups

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