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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
MathSciNet review: 3767368
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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
MR Author ID: 197703
Email: astash56@mail.ru

Lech Maligranda
Affiliation: Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
MR Author ID: 118770
Email: lech.maligranda@ltu.se

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