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The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B


Authors: Richard Aron, Yun Sung Choi, Sun Kwang Kim, Han Ju Lee and Miguel Martín
Journal: Trans. Amer. Math. Soc. 367 (2015), 6085-6101
MSC (2010): Primary 46B20; Secondary 46B04, 46B22
DOI: https://doi.org/10.1090/S0002-9947-2015-06551-9
Published electronically: March 2, 2015
MathSciNet review: 3356930
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Abstract: We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces $ X$ such that $ (X,Y)$ has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space $ Y$. We show that in this case, there exists a universal function $ \eta _X(\varepsilon )$ such that for every $ Y$, the pair $ (X,Y)$ has the BPBp with this function. This allows us to prove some necessary isometric conditions for $ X$ to have the property. We also prove that if $ X$ has this property in every equivalent norm, then $ X$ is one-dimensional. For range spaces, we study Banach spaces $ Y$ such that $ (X,Y)$ has the Bishop-Phelps-Bollobás property for every Banach space $ X$. In this case, we show that there is a universal function $ \eta _Y(\varepsilon )$ such that for every $ X$, the pair $ (X,Y)$ has the BPBp with this function. This implies that this property of $ Y$ is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for $ c_0$-, $ \ell _1$- and $ \ell _\infty $-sums of Banach spaces.


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Additional Information

Richard Aron
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: aron@math.kent.edu

Yun Sung Choi
Affiliation: Department of Mathematics, POSTECH, Pohang (790-784), Republic of Korea
Email: mathchoi@postech.ac.kr

Sun Kwang Kim
Affiliation: Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea
Email: sunkwang@kgu.ac.kr

Han Ju Lee
Affiliation: Department of Mathematics Education, Dongguk University - Seoul, 100-715 Seoul, Republic of Korea
Email: hanjulee@dongguk.edu

Miguel Martín
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain
Email: mmartins@ugr.es

DOI: https://doi.org/10.1090/S0002-9947-2015-06551-9
Keywords: Banach space, approximation, norm-attaining operators, Bishop-Phelps-Bollob\'as theorem.
Received by editor(s): May 28, 2013
Published electronically: March 2, 2015
Additional Notes: The first author was partially supported by Spanish MICINN Project MTM2011-22417. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2010-0008543 and No. 2013053914). The third author was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2014R1A1A2056084). The fourth author was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A1006869). The fifth author was partially supported by Spanish MICINN and FEDER project no. MTM2012-31755, Junta de Andalucía and FEDER grants FQM-185 and P09-FQM-4911, and by “Programa Nacional de Movilidad de Recursos Humanos del Plan Nacional de I+D+i 2008–2011” of the Spanish MECD
Dedicated: Dedicated to the memory of Joram Lindenstrauss and Robert Phelps
Article copyright: © Copyright 2015 American Mathematical Society

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