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The Daugavet property in rearrangement invariant spaces


Authors: M. D. Acosta, A. Kamińska and M. Mastyło
Journal: Trans. Amer. Math. Soc. 367 (2015), 4061-4078
MSC (2010): Primary 46B20, 46E30
Published electronically: December 3, 2014
MathSciNet review: 3324920
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Abstract: We study rearrangement invariant spaces with the Daugavet property. The main result of this paper states that under mild assumptions the only nonseparable rearrangement invariant space $ X$ over an atomless finite measure space with the Daugavet property is $ L_{\infty }$ endowed with its canonical norm. We also prove that a uniformly monotone rearrangement invariant space over an infinite atomless measure space with the Daugavet property is isometric to $ L_1$. As an application we obtain that an Orlicz space over an atomless measure space has the Daugavet property if and only if it is isometrically isomorphic to $ L_1$.


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Additional Information

M. D. Acosta
Affiliation: Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain
Email: dacosta@ugr.es

A. Kamińska
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email: kaminska@memphis.edu

M. Mastyło
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University and Institute of Mathematics, Polish Academy of Sciences (Poznań branch), Umultowska 87, 61-614 Poznań, Poland
Email: mastylo@amu.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-2014-06166-7
Keywords: Daugavet property, rearrangement invariant spaces, Orlicz spaces, uniform monotonicity
Received by editor(s): November 29, 2012
Received by editor(s) in revised form: March 9, 2013
Published electronically: December 3, 2014
Additional Notes: The first author was supported by MTM2012-31755, Junta de Andalucía FQM–4911 and FQM–185.
The third author was supported by the National Science Centre (NCN), Poland, grant no. 2011/01/B/ST1/06243.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.