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DAUGAVET PROPERTY IN TENSOR PRODUCT SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  21 November 2019

Abraham Rueda Zoca Open the ORCID record for Abraham Rueda Zoca [Opens in a new window] ,
Pedro Tradacete Open the ORCID record for Pedro Tradacete [Opens in a new window]  and
Ignacio Villanueva
Abraham Rueda Zoca
Affiliation:
Universidad de Granada, Facultad de Ciencias, Departamento de Análisis Matemático, 18071-Granada, Spain (abrahamrueda@.ugr.es), URL: https://arzenglish.wordpress.com
Pedro Tradacete
Affiliation:
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13–15, Campus de Cantoblanco UAM, 28049Madrid, Spain (pedro.tradacete@icmat.es), URL: https://www.icmat.es/miembros/ptradacete/
Ignacio Villanueva
Affiliation:
Universidad Complutense de Madrid, Departamento de Análisis Matemtico y Matemática Aplicada, Instituto de Matemática Interdisciplinar-IMI, Instituto de Ciencias Matemáticas ICMAT, Madrid, Spain (ignaciov@ucm.es)
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Abstract

We study the Daugavet property in tensor products of Banach spaces. We show that $L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$ has the Daugavet property when $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$ are purely non-atomic measures. Also, we show that $X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$ has the Daugavet property provided $X$ and $Y$ are $L_{1}$-preduals with the Daugavet property, in particular, spaces of continuous functions with this property. With the same techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.

Keywords

Daugavet propertytensor product spacesoctahedral norms

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces 46B28: Spaces of operators; tensor products; approximation properties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 4 , July 2021 , pp. 1409 - 1428
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Copyright
© The Author(s), 2019. Published by Cambridge University Press

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Footnotes

The research of the first author was supported by MECD (Spain) FPU2016/00015, MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE), Junta de Andalucía Grant A-FQM-484-UGR18 and Junta de Andalucía Grant FQM-0185. The second author gratefully acknowledges the support of MINECO (Spain) through grants MTM2016-76808-P (AEI/FEDER, UE) and MTM2016-75196-P (AEI/FEDER, UE) and the ‘Severo Ochoa Programme for Centres of Excellence in R&D’ (SEV-2015-0554). The third author was supported by MINECO (Spain) Grant MTM2017-88385-P, QUITEMAD+-CM (S2013/ICE- 2801) and the ‘Severo Ochoa Programme for Centres of Excellence in R&D’ (SEV-2015-0554).

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