## On spreading sequences and asymptotic structures

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- by D. Freeman, E. Odell, B. Sarı and B. Zheng PDF
- Trans. Amer. Math. Soc.
**370**(2018), 6933-6953 Request permission

## Abstract:

In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibits a striking resemblance to the geometry of James space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal.

The second part contains two results on Banach spaces $X$ whose asymptotic structures are closely related to $c_0$ and do not contain a copy of $\ell _1$:

i) Suppose $X$ has a normalized weakly null basis $(x_i)$ and every spreading model $(e_i)$ of a normalized weakly null block basis satisfies $\|e_1-e_2\|=1$. Then some subsequence of $(x_i)$ is equivalent to the unit vector basis of $c_0$. This generalizes a similar theorem of Odell and Schlumprecht and yields a new proof of the Elton–Odell theorem on the existence of infinite $(1+\varepsilon )$-separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space.

ii) Suppose that all asymptotic models of $X$ generated by weakly null arrays are equivalent to the unit vector basis of $c_0$. Then $X^*$ is separable and $X$ is asymptotic-$c_0$ with respect to a shrinking basis $(y_i)$ of $Y\supseteq X$.

## References

- Spiros A. Argyros and Richard G. Haydon,
*A hereditarily indecomposable $\scr L_\infty$-space that solves the scalar-plus-compact problem*, Acta Math.**206**(2011), no. 1, 1–54. MR**2784662**, DOI 10.1007/s11511-011-0058-y - Dale Alspach, Robert Judd, and Edward Odell,
*The Szlenk index and local $l_1$-indices*, Positivity**9**(2005), no. 1, 1–44. MR**2139115**, DOI 10.1007/s11117-002-9781-0 - Spiros A. Argyros and Pavlos Motakis,
*A hereditarily indecomposable Banach space with rich spreading model structure*, Israel J. Math.**203**(2014), no. 1, 341–387. MR**3273444**, DOI 10.1007/s11856-014-1099-7 - Spiros A. Argyros, Pavlos Motakis, and Bünyamin Sarı,
*A study of conditional spreading sequences*, J. Funct. Anal.**273**(2017), no. 3, 1205–1257. MR**3653952**, DOI 10.1016/j.jfa.2017.04.009 - J. Bourgain and F. Delbaen,
*A class of special ${\cal L}_{\infty }$ spaces*, Acta Math.**145**(1980), no. 3-4, 155–176. MR**590288**, DOI 10.1007/BF02414188 - Steven F. Bellenot, Richard Haydon, and Edward Odell,
*Quasi-reflexive and tree spaces constructed in the spirit of R. C. James*, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 19–43. MR**983379**, DOI 10.1090/conm/085/983379 - C. Bessaga and A. Pełczyński,
*On subspaces of a space with an absolute basis*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.**6**(1958), 313–315. MR**0096956** - Antoine Brunel and Louis Sucheston,
*Equal signs additive sequences in Banach spaces*, J. Functional Analysis**21**(1976), no. 3, 286–304. MR**0397369**, DOI 10.1016/0022-1236(76)90041-0 - John Hancock Elton,
*WEAKLY NULL NORMALIZED SEQUENCES IN BANACH SPACES*, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–Yale University. MR**2628434** - J. Elton and E. Odell,
*The unit ball of every infinite-dimensional normed linear space contains a $(1+\varepsilon )$-separated sequence*, Colloq. Math.**44**(1981), no. 1, 105–109. MR**633103**, DOI 10.4064/cm-44-1-105-109 - D. Freeman, E. Odell, and Th. Schlumprecht,
*The universality of $\ell _1$ as a dual space*, Math. Ann.**351**(2011), no. 1, 149–186. MR**2824850**, DOI 10.1007/s00208-010-0601-8 - James Hagler,
*A counterexample to several questions about Banach spaces*, Studia Math.**60**(1977), no. 3, 289–308. MR**442651**, DOI 10.4064/sm-60-3-289-308 - Lorenz Halbeisen and Edward Odell,
*On asymptotic models in Banach spaces*, Israel J. Math.**139**(2004), 253–291. MR**2041794**, DOI 10.1007/BF02787552 - Joram Lindenstrauss and Lior Tzafriri,
*Classical Banach spaces. I*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR**0500056**, DOI 10.1007/978-3-642-66557-8 - B. Maurey, V. D. Milman, and N. Tomczak-Jaegermann,
*Asymptotic infinite-dimensional theory of Banach spaces*, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 149–175. MR**1353458** - E. Odell,
*Stability in Banach spaces*, Extracta Math.**17**(2002), no. 3, 385–425. IV Course on Banach Spaces and Operators (Spanish) (Laredo, 2001). MR**1995414** - E. Odell and Th. Schlumprecht,
*A problem on spreading models*, J. Funct. Anal.**153**(1998), no. 2, 249–261. MR**1614578**, DOI 10.1006/jfan.1997.3191 - Haskell P. Rosenthal,
*A characterization of Banach spaces containing $l^{1}$*, Proc. Nat. Acad. Sci. U.S.A.**71**(1974), 2411–2413. MR**358307**, DOI 10.1073/pnas.71.6.2411 - Haskell Rosenthal,
*A characterization of Banach spaces containing $c_0$*, J. Amer. Math. Soc.**7**(1994), no. 3, 707–748. MR**1242455**, DOI 10.1090/S0894-0347-1994-1242455-4 - Charles Stegall,
*The Radon-Nikodym property in conjugate Banach spaces*, Trans. Amer. Math. Soc.**206**(1975), 213–223. MR**374381**, DOI 10.1090/S0002-9947-1975-0374381-1

## Additional Information

**D. Freeman**- Affiliation: Department of Mathematics and Computer Science, Saint Louis University , St. Louis, Missouri 63103
- MR Author ID: 742577
- Email: dfreema7@slu.edu
**E. Odell**- Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712-0257
**B. Sarı**- Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203-5017
- MR Author ID: 741208
- Email: bunyamin@unt.edu
**B. Zheng**- Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152-3240
- Email: bzheng@memphis.edu
- Received by editor(s): July 18, 2016
- Received by editor(s) in revised form: January 16, 2017
- Published electronically: April 4, 2018
- Additional Notes: Research of the first, second, and fourth authors was supported by the National Science Foundation.

Research of the first author was also supported by grant 353293 from the Simons Foundation.

Edward Odell (1947–2013).

The third author was supported by grant 208290 from the Simons Foundation.

The fourth author is the corresponding author. His research was supported in part by the National Science Foundation of China grant 11628102. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 6933-6953 - MSC (2010): Primary 46B03, 46B25, 46B45, 46B06; Secondary 05D10
- DOI: https://doi.org/10.1090/tran/7189
- MathSciNet review: 3841837