The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B
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- by Richard Aron, Yun Sung Choi, Sun Kwang Kim, Han Ju Lee and Miguel Martín PDF
- Trans. Amer. Math. Soc. 367 (2015), 6085-6101 Request permission
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.References
- María D. Acosta, Denseness of norm attaining mappings, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 9–30 (English, with English and Spanish summaries). MR 2267397
- María D. Acosta, Richard M. Aron, Domingo García, and Manuel Maestre, The Bishop-Phelps-Bollobás theorem for operators, J. Funct. Anal. 254 (2008), no. 11, 2780–2799. MR 2414220, DOI 10.1016/j.jfa.2008.02.014
- María D. Acosta, Julio Becerra-Guerrero, Domingo García, and Manuel Maestre, The Bishop-Phelps-Bollobás theorem for bilinear forms, Trans. Amer. Math. Soc. 365 (2013), no. 11, 5911–5932. MR 3091270, DOI 10.1090/S0002-9947-2013-05881-3
- María D. Acosta, Francisco J. Aguirre, and Rafael Payá, A new sufficient condition for the denseness of norm attaining operators, Rocky Mountain J. Math. 26 (1996), no. 2, 407–418. MR 1406488, DOI 10.1216/rmjm/1181072066
- Richard M. Aron, Yun Sung Choi, Domingo García, and Manuel Maestre, The Bishop-Phelps-Bollobás theorem for $\scr L(L_1(\mu ), L_\infty [0,1])$, Adv. Math. 228 (2011), no. 1, 617–628. MR 2822241, DOI 10.1016/j.aim.2011.05.023
- Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. MR 123174, DOI 10.1090/S0002-9904-1961-10514-4
- Béla Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181–182. MR 267380, DOI 10.1112/blms/2.2.181
- J. Bourgain, On dentability and the Bishop-Phelps property, Israel J. Math. 28 (1977), no. 4, 265–271. MR 482076, DOI 10.1007/BF02760634
- B. Cascales, A. J. Guirao, and V. Kadets, A Bishop-Phelps-Bollobás type theorem for uniform algebras, Adv. Math. 240 (2013), 370–382. MR 3046314, DOI 10.1016/j.aim.2013.03.005
- Yun Sung Choi and Sun Kwang Kim, The Bishop-Phelps-Bollobás property and lush spaces, J. Math. Anal. Appl. 390 (2012), no. 2, 549–555. MR 2890536, DOI 10.1016/j.jmaa.2012.01.053
- Yun Sung Choi and Sun Kwang Kim, The Bishop-Phelps-Bollobás theorem for operators from $L_1(\mu )$ to Banach spaces with the Radon-Nikodým property, J. Funct. Anal. 261 (2011), no. 6, 1446–1456. MR 2813477, DOI 10.1016/j.jfa.2011.05.007
- Yun Sung Choi and Hyun Gwi Song, Property (quasi-$\alpha$) and the denseness of norm attaining mappings, Math. Nachr. 281 (2008), no. 9, 1264–1272. MR 2442704, DOI 10.1002/mana.200510676
- Yun Sung Choi and Hyun Gwi Song, The Bishop-Phelps-Bollobás theorem fails for bilinear forms on $l_1\times l_1$, J. Math. Anal. Appl. 360 (2009), no. 2, 752–753. MR 2561271, DOI 10.1016/j.jmaa.2009.07.008
- Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. The basis for linear and nonlinear analysis. MR 2766381, DOI 10.1007/978-1-4419-7515-7
- G. J. O. Jameson, Topology and normed spaces, Chapman and Hall, London; Halsted Press [John Wiley & Sons, Inc.], New York, 1974. MR 0463890
- Sun Kwang Kim, The Bishop-Phelps-Bollobás theorem for operators from $c_0$ to uniformly convex spaces, Israel J. Math. 197 (2013), no. 1, 425–435. MR 3096622, DOI 10.1007/s11856-012-0186-x
- Sun Kwang Kim and Han Ju Lee, Uniform convexity and Bishop-Phelps-Bollobás property, Canad. J. Math. 66 (2014), no. 2, 373–386. MR 3176146, DOI 10.4153/CJM-2013-009-2
- Joram Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139–148. MR 160094, DOI 10.1007/BF02759700
- Rafael Payá and Yousef Saleh, Norm attaining operators from $L_1(\mu )$ into $L_\infty (\nu )$, Arch. Math. (Basel) 75 (2000), no. 5, 380–388. MR 1785447, DOI 10.1007/s000130050519
- J. R. Partington, Norm attaining operators, Israel J. Math. 43 (1982), no. 3, 273–276. MR 689984, DOI 10.1007/BF02761947
- Walter Schachermayer, Norm attaining operators and renormings of Banach spaces, Israel J. Math. 44 (1983), no. 3, 201–212. MR 693659, DOI 10.1007/BF02760971
- Walter Schachermayer, Norm attaining operators on some classical Banach spaces, Pacific J. Math. 105 (1983), no. 2, 427–438. MR 691613
Additional Information
- Richard Aron
- Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
- MR Author ID: 27325
- 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
- MR Author ID: 643000
- ORCID: 0000-0003-4502-798X
- Email: mmartins@ugr.es
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3356930
Dedicated: Dedicated to the memory of Joram Lindenstrauss and Robert Phelps