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Non-isomorphic complemented subspaces of the reflexive Orlicz function spaces $ L^{\Phi}[0,1]$


Author: Ghadeer Ghawadrah
Journal: Proc. Amer. Math. Soc. 144 (2016), 285-299
MSC (2010): Primary 46B20; Secondary 54H05
DOI: https://doi.org/10.1090/proc12712
Published electronically: May 28, 2015
MathSciNet review: 3415596
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Abstract: In this note we show that the number of isomorphism classes of complemented subspaces of a reflexive Orlicz function space $ L^{\Phi }[0,1]$ is uncountable, as soon as $ L^{\Phi }[0,1]$ is not isomorphic to $ L^{2}[0,1]$. Also, we prove that the set of all separable Banach spaces that are isomorphic to such an $ L^{\Phi }[0,1]$ is analytic non-Borel. Moreover, by using the Boyd interpolation theorem we extend some results on $ L^{p}[0,1]$ spaces to the rearrangement invariant function spaces under natural conditions on their Boyd indices.


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  • [AGR03] Spiros A. Argyros, Gilles Godefroy, and Haskell P. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1007-1069. MR 1999190 (2004g:46002), https://doi.org/10.1016/S1874-5849(03)80030-X
  • [ASW11] Sergey Astashkin, Fedor Sukochev, and Chin Pin Wong, Disjointification of martingale differences and conditionally independent random variables with some applications, Studia Math. 205 (2011), no. 2, 171-200. MR 2824894 (2012h:60134), https://doi.org/10.4064/sm205-2-3
  • [BG70] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304. MR 0440695 (55 #13567)
  • [Bos02] Benoît Bossard, A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fund. Math. 172 (2002), no. 2, 117-152. MR 1899225 (2003d:46016), https://doi.org/10.4064/fm172-2-3
  • [Bou81] Jean Bourgain, New classes of $ {\mathcal {L}}^{p}$-spaces, Lecture Notes in Mathematics, vol. 889, Springer-Verlag, Berlin-New York, 1981. MR 639014 (83j:46028)
  • [Boy69] David W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245-1254. MR 0412788 (54 #909)
  • [BP58] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 0115069 (22 #5872)
  • [BRS81] J. Bourgain, H. P. Rosenthal, and G. Schechtman, An ordinal $ L^{p}$-index for Banach spaces, with application to complemented subspaces of $ L^{p}$, Ann. of Math. (2) 114 (1981), no. 2, 193-228. MR 632839 (83j:46031), https://doi.org/10.2307/1971293
  • [DK14] S. Dutta and D. Khurana, Ordinal indices for complemented subspaces of $ l_{p}$, arXiv preprint arXiv:1405.4499 (2014).
  • [GG73] J. L. B. Gamlen and R. J. Gaudet, On subsequences of the Haar system in $ L_{p}$ $ [1,\,1](1\leq p\leq \infty )$, Israel J. Math. 15 (1973), 404-413. MR 0328575 (48 #6917)
  • [God10] Gilles Godefroy, Analytic sets of Banach spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 104 (2010), no. 2, 365-374 (English, with English and Spanish summaries). MR 2757247 (2011m:46008), https://doi.org/10.5052/RACSAM.2010.23
  • [HP86] Francisco L. Hernández and Vicente Peirats, Orlicz function spaces without complemented copies of $ l^p$, Israel J. Math. 56 (1986), no. 3, 355-360. MR 882259 (88a:46028), https://doi.org/10.1007/BF02782943
  • [HR89] Francisco L. Hernández and Baltasar Rodríguez-Salinas, On minimality and $ l^p$-complemented subspaces of Orlicz function spaces, Rev. Mat. Univ. Complut. Madrid 2 (1989), no. suppl., 129-136. Congress on Functional Analysis (Madrid, 1988). MR 1057214 (91k:46024)
  • [JMST79] W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010 (82j:46025), https://doi.org/10.1090/memo/0217
  • [JS89] William B. Johnson and G. Schechtman, Sums of independent random variables in rearrangement invariant function spaces, Ann. Probab. 17 (1989), no. 2, 789-808. MR 985390 (90h:60045)
  • [Kal93] N. J. Kalton, $ M$-ideals of compact operators, Illinois J. Math. 37 (1993), no. 1, 147-169. MR 1193134 (94b:46028)
  • [Kec95] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • [KP98] Paweł Kolwicz and Ryszard Płuciennik, $ P$-convexity of Orlicz-Bochner spaces, Proc. Amer. Math. Soc. 126 (1998), no. 8, 2315-2322. MR 1443391 (98j:46032), https://doi.org/10.1090/S0002-9939-98-04290-7
  • [KW95] Nigel J. Kalton and Dirk Werner, Property $ (M)$, $ M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137-178. MR 1324212 (96m:46022), https://doi.org/10.1515/crll.1995.461.137
  • [LT79] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367 (81c:46001)
  • [Rui91] César Ruiz, On subspaces of Orlicz function spaces spanned by sequences of independent symmetric random variables, Function spaces (Poznań, 1989) Teubner-Texte Math., vol. 120, Teubner, Stuttgart, 1991, pp. 41-48. MR 1155156 (93b:46050)

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Additional Information

Ghadeer Ghawadrah
Affiliation: Université Paris VI, Institut de Mathématiques de Jussieu, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France
Email: ghadeer.ghawadrah@imj-prg.fr

DOI: https://doi.org/10.1090/proc12712
Keywords: Orlicz function space, complemented subspaces, Cantor group, rearrangement invariant function space, well-founded tree, analytic, non-Borel
Received by editor(s): August 9, 2014
Received by editor(s) in revised form: October 23, 2014, December 19, 2014, and December 20, 2014
Published electronically: May 28, 2015
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society

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