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Transactions of the American Mathematical Society

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On spreading sequences and asymptotic structures


Authors: D. Freeman, E. Odell, B. Sarı and B. Zheng
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
Published electronically: April 4, 2018
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Abstract | References | Similar Articles | Additional Information

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 $ \Vert e_1-e_2\Vert=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$.


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Additional Information

D. Freeman
Affiliation: Department of Mathematics and Computer Science, Saint Louis University , St. Louis, Missouri 63103
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
Email: bunyamin@unt.edu

B. Zheng
Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152-3240
Email: bzheng@memphis.edu

DOI: https://doi.org/10.1090/tran/7189
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.
Article copyright: © Copyright 2018 American Mathematical Society

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