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$ L_p +L_{\infty}$ and $ L_p \cap L_{\infty}$ are not isomorphic for all $ 1 \leq p < \infty$, $ p \neq 2$


Authors: Sergey V. Astashkin and Lech Maligranda
Journal: Proc. Amer. Math. Soc. 146 (2018), 2181-2194
MSC (2010): Primary 46E30, 46B20, 46B42
DOI: https://doi.org/10.1090/proc/13928
Published electronically: February 1, 2018
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Abstract: We prove the result stated in the title. It comes as a consequence of the fact that the space $ L_p \cap L_{\infty }$, $ 1\leq p<\infty $, $ p\neq 2$, does not contain a complemented subspace isomorphic to $ L_p$. In particular, as a subproduct, we show that $ L_p \cap L_{\infty }$ contains a complemented subspace isomorphic to $ {\ell }_2$ if and only if $ p = 2$.


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Additional Information

Sergey V. Astashkin
Affiliation: Department of Mathematics Samara National Research University, Moskovskoye shosse 34 443086, Samara, Russia
Email: astash56@mail.ru

Lech Maligranda
Affiliation: Department of Engineering Sciences and Mathematics Luleå University of Technology SE-971 87 Luleå, Sweden
Email: lech.maligranda@ltu.se

DOI: https://doi.org/10.1090/proc/13928
Keywords: Symmetric spaces, isomorphic spaces, complemented subspaces
Received by editor(s): June 23, 2017
Received by editor(s) in revised form: August 10, 2017
Published electronically: February 1, 2018
Additional Notes: The research of the first author was partially supported by the Ministry of Education and Science of the Russian Federation, project 1.470.2016/1.4, and by the RFBR grant 17-01-00138.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2018 American Mathematical Society

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