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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Complemented isometric copies of $L_{1}$ in dual Banach spaces

Author: J. Hagler
Journal: Proc. Amer. Math. Soc. 130 (2002), 3313-3324
MSC (2000): Primary 46B04, 46B10; Secondary 46B20
Published electronically: March 25, 2002
MathSciNet review: 1913011
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $X$ be a real or complex Banach space and $K\geq1$. Then $X^{\ast}$contains a $K$-complemented, isometric copy of $L_{1}\left[ 0,1\right] $ if and only if $X^{\ast}$ contains a $K$-complemented, isometric copy of $C\left[0,1\right] ^{\ast}$ if and only if $X$ contains a subspace $\left( 1,K\right) $-asymptotic to $\left( \ell_{1}\oplus\sum_{n}\ell_{\infty} ^{n}\right)_{1}$.

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  • [DGH] S. Dilworth, M. Girardi and J. Hagler, Dual Banach spaces which contain an isometric copy of $L_{1}$, Bull. Polon. Acad. Sci. 48 (2000), 1-12. MR 2001e:46016
  • [DJLT] P. N. Dowling, W. B. Johnson, C. J. Lennard and B. Turett, The optimality of James's distortion theorems, Proc. Amer. Math. Soc. 125 (1997), 167-174. MR 97d:46010
  • [DL] P. N. Dowling and C. J. Lennard, Every nonreflexive subspace of $L_{1}$ fails the fixed point property, Proc. Amer. Math. Soc. 125 (1997), 443-446. MR 97d:46034
  • [DLT] P. N. Dowling, C. J. Lennard and B. Turett, Reflexivity and the fixed-point property for neonexpansive maps, J. Math. Analysis and Applications 200 (1996), 653-662. MR 97c:47062
  • [DRT] Patrick N. Dowling, Narcisse Randrianantoanina and Barry Turett, Remarks on James's distortion theorems, Bull. Austral. Math. Soc. 57 (1998), 49-54. MR 99b:46014
  • [H1] J. Hagler, Embeddings of $L^{1}$ into conjugate Banach spaces, Ph.D. Thesis, University of California, Berkeley, Calif., 1972.
  • [H2] J. Hagler, Some more Banach spaces which contain $\ell^{1}$, Studia Math. 46 (1973), 35-42. MR 48:11995
  • [HS] J. Hagler and C. Stegall, Banach spaces whose duals contain complemented subspaces isomorphic to $C[0,1]^{\ast}$, J. Funct. Anal.13 (1973), 233-251. MR 50:2874
  • [J] W. B. Johnson, A complementably universal conjugate Banach space and its relation to the approximation property,Israel J. Math. 13 (1972), 301-310. MR 48:4700
  • [JRZ] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. MR 43:6702
  • [L] J. Lindenstrauss, A short proof of Liapounoff's convexity theorem, J. Math. and Mech. 15 (1966), 971-972. MR 34:7754
  • [LR] J. Lindenstrauss and H. P. Rosenthal, The $\mathcal{L}_{p}$spaces, Israel J. Math. 7 (1969), 325-349. MR 42:5012
  • [P] A. Pe\lczynski, On Banach spaces containing $L_{1}(\mu)$, Studia Math. 30 (1968), 231-246. MR 38:521
  • [R] H. P. Rosenthal, On factors of $C([0,1])$ with non-separable dual, Israel J. Math. 13 (1972), 361-378. MR 52:8900
  • [S] C. Stegall, Banach spaces whose duals contain $\ell_{1}\left( \Gamma\right) $ with applications to the study of dual $L_{1}\left( \mu\right) $ spaces, Trans. Amer. Math. Soc. 176 (1993), 463-477. MR 47:3953

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Additional Information

J. Hagler
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208

PII: S 0002-9939(02)06474-2
Keywords: Banach spaces, complemented isometric copies of $L_1$, $\left( 1,K\right) $\emph{-}asymptotic copies of\emph{ }$\left( \ell_{1}\oplus\sum_{n}\ell_{\infty}^{n}\right) _{1}$
Received by editor(s): January 30, 2001
Received by editor(s) in revised form: June 13, 2001
Published electronically: March 25, 2002
Additional Notes: The author would especially like to thank H. P. Rosenthal and C. Stegall. Thanks also go to M. Girardi, S. Dilworth, W. B. Johnson and the referee for helpful comments and suggestions
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society