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NONCOMMUTATIVE DE LEEUW THEOREMS

Part of: Selfadjoint operator algebras Harmonic analysis in several variables Locally compact groups and their algebras

Published online by Cambridge University Press:  12 October 2015

MARTIJN CASPERS ,
JAVIER PARCET ,
MATHILDE PERRIN  and
ÉRIC RICARD
MARTIJN CASPERS
Affiliation:
Fachbereich Mathematik und Informatik, Westfälische Wilhelmsuniversität Münster, Einsteinstrasse 62, 48149 Münster, Germany; martijn.caspers@uni-muenster.de
JAVIER PARCET
Affiliation:
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/Nicolás Cabrera 13-15, 28049 Madrid, Spain; javier.parcet@icmat.es, mathilde.perrin@icmat.es
MATHILDE PERRIN
Affiliation:
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/Nicolás Cabrera 13-15, 28049 Madrid, Spain; javier.parcet@icmat.es, mathilde.perrin@icmat.es
ÉRIC RICARD
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Normandie, 14032 Caen Cedex, France; eric.ricard@unicaen.fr

Abstract

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Let $\text{H}$ be a subgroup of some locally compact group $\text{G}$. Assume that $\text{H}$ is approximable by discrete subgroups and that $\text{G}$ admits neighborhood bases which are almost invariant under conjugation by finite subsets of $\text{H}$. Let $m:\text{G}\rightarrow \mathbb{C}$ be a bounded continuous symbol giving rise to an $L_{p}$-bounded Fourier multiplier (not necessarily completely bounded) on the group von Neumann algebra of $\text{G}$ for some $1\leqslant p\leqslant \infty$. Then, $m_{\mid _{\text{H}}}$ yields an $L_{p}$-bounded Fourier multiplier on the group von Neumann algebra of $\text{H}$ provided that the modular function ${\rm\Delta}_{\text{G}}$ is equal to 1 over $\text{H}$. This is a noncommutative form of de Leeuw’s restriction theorem for a large class of pairs $(\text{G},\text{H})$. Our assumptions on $\text{H}$ are quite natural, and they recover the classical result. The main difference with de Leeuw’s original proof is that we replace dilations of Gaussians by other approximations of the identity for which certain new estimates on almost-multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.

MSC classification

Primary: 42B15: Multipliers
Secondary: 22D15: Group algebras of locally compact groups 46L52: Noncommutative function spaces
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e21
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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