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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Schur and Fourier multipliers of an amenable group acting on non-commutative $L^p$-spaces


Authors: Martijn Caspers and Mikael de la Salle
Journal: Trans. Amer. Math. Soc. 367 (2015), 6997-7013
MSC (2010): Primary 43A15, 46B08, 46B28, 46B70
DOI: https://doi.org/10.1090/S0002-9947-2015-06281-3
Published electronically: March 4, 2015
MathSciNet review: 3378821
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Abstract:

Consider a completely bounded Fourier multiplier $\phi$ of a locally compact group $G$, and take $1 \leq p \leq \infty$. One can associate to $\phi$ a Schur multiplier on the Schatten classes $\mathcal {S}_p(L^2 G)$, as well as a Fourier multiplier on $L^p(\mathcal {L} G)$, the non-commutative $L^p$-space of the group von Neumann algebra of $G$. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the $L^p$-Fourier multiplier. When $G$ is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups.

For a discrete group $G$ and in the special case when $p\neq 2$ is an even integer, we show the following. If there exists a map between $L^p(\mathcal {L} G)$ and an ultraproduct of $L^p(\mathcal {M}) \otimes \mathcal {S}_p(L^2G)$ that intertwines the Fourier multiplier with the Schur multiplier, then $G$ must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups.


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Additional Information

Martijn Caspers
Affiliation: Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France
Address at time of publication: Einsteinstrasse 62, D-48149 Münster, Germany
Email: martijn.caspers@univ-fcomte.fr, martijn.caspers@uni-muenster.de

Mikael de la Salle
Affiliation: CNRS, Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France
Address at time of publication: ENS de Lyon, (site Sciences), 46, allée d’Italie, 69364 Lyon Cedex 07, France
Email: mikael.de_la_salle@univ-fcomte.fr, mikael.de.la.salle@ens-lyon.fr

Keywords: Non-commutative $L^p$-spaces, Schur multiplier, Fourier multipliers, amenability
Received by editor(s): March 5, 2013
Received by editor(s) in revised form: June 18, 2013
Published electronically: March 4, 2015
Additional Notes: The first author was supported by the ANR project ANR-2011-BS01-008-01
The second author was partially supported by the ANR projects NEUMANN and OSQPI
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.