## Duke Mathematical Journal

### A characterization of subspaces and quotients of reflexive banach spaces with unconditional bases

#### Abstract

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property (UTP) has the UTP. This is used to prove that a separable reflexive Banach space with the UTP embeds into a reflexive Banach space with an unconditional basis. This solves several longstanding open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite-dimensional decomposition (UFDD) embeds into a reflexive Banach space with an unconditional basis

#### Article information

Source
Duke Math. J., Volume 141, Number 3 (2008), 505-518.

Dates
First available in Project Euclid: 15 February 2008

https://projecteuclid.org/euclid.dmj/1203087635

Digital Object Identifier
doi:10.1215/00127094-2007-003

Mathematical Reviews number (MathSciNet)
MR2387429

Zentralblatt MATH identifier
1146.46003

#### Citation

Johnson, W. B.; Zheng, Bentuo. A characterization of subspaces and quotients of reflexive banach spaces with unconditional bases. Duke Math. J. 141 (2008), no. 3, 505--518. doi:10.1215/00127094-2007-003. https://projecteuclid.org/euclid.dmj/1203087635

#### References

• W. J. Davis, T. Figiel, W. B. Johnson, and A. PełCzyńSki, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311--327.
• M. Fabian, P. Habala, P. HáJek, V. Montesinos SantalucíA, J. Pelant, and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books Math./Ouvrages Math. SMC 8, Springer, New York, 2001.
• M. Feder, On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math. 24 (1980), 196--205.
• T. Figiel, W. B. Johnson, and L. Tzafriri, On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces, J. Approximation Theory 13 (1975), 395--412.
• R. Haydon, E. Odell, and T. Schlumprecht, Small subspaces of $L_p$, preprint, \arxiv0711.3919v1 [math.FA]
• W. B. Johnson, On quotients of $L_p$ which are quotients of $\ell_p$, Compositio Math. 34 (1977), 69--89.
• W. B. Johnson and H. P. Rosenthal, On $\omega^*$-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77--92.
• W. B. Johnson and A. Szankowski, Complementably universal Banach spaces, Studia Math. 58 (1976), 91--97.
• W. B. Johnson and M. Zippin, On subspaces of quotients of $\big(\sumG_n\big)_l_p$ and $\big(\sumG_n\big)_c_0$'' in Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normal Linear Spaces (Jerusalem, 1972), Israel J. Math. 13 (1972), 311--316.
• —, Subspaces and quotient spaces of $\big(\sumG_n\big)_l_p$ and $\big(\sumG_n\big)_c_0$, Israel J. Math. 17 (1974), 50--55.
• J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I: Sequence Spaces, Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977.
• E. Odell and T. Schlumprecht, Trees and branches in Banach spaces, Trans. Amer. Math. Soc. 354, no. 10 (2002), 4085--4108.
• —, A universal reflexive space for the class of uniformly convex Banach spaces, Math. Ann. 335 (2006), 901--916.
• E. Odell, T. Schlumprecht, and A. ZsáK, On the structure of asymptotic $\ell_p$ spaces, to appear in Q. J. Math.
• A. PełCzyńSki and P. Wojtaszczyk, Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces, Studia Math. 40 (1971), 91--108.
• T. Schlumprecht, private communication, 2006.
• M. Zippin, Banach spaces with separable duals, Trans. Amer. Math. Soc. 310, no. 1 (1988), 371--379.