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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
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

DOI: https://doi.org/10.1090/S0002-9947-2015-06281-3
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.