Unconditional structures of weakly null sequences

Argyros, S.; Gasparis, I.

doi:10.1090/S0002-9947-01-02711-8

Unconditional structures of weakly null sequences
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by S. A. Argyros and I. Gasparis

Trans. Amer. Math. Soc. 353 (2001), 2019-2058

DOI: https://doi.org/10.1090/S0002-9947-01-02711-8

Published electronically: January 10, 2001

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Abstract:

The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of $c_0$, or there exists a subsequence which is boundedly convexly complete.

References

Dale E. Alspach and Spiros Argyros, Complexity of weakly null sequences, Dissertationes Math. (Rozprawy Mat.) 321 (1992), 44. MR 1191024
D. Alspach and E. Odell, Averaging weakly null sequences, Functional analysis (Austin, TX, 1986–87) Lecture Notes in Math., vol. 1332, Springer, Berlin, 1988, pp. 126–144. MR 967092, DOI 10.1007/BFb0081615
G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125–149. MR 1679593, DOI 10.1007/BF02775031
S. A. Argyros, S. Mercourakis, and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157–193. MR 1658551, DOI 10.1007/BF02764008
S.A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000), 243–294.
C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
Pilar Cembranos, The hereditary Dunford-Pettis property on $C(K,E)$, Illinois J. Math. 31 (1987), no. 3, 365–373. MR 892175
J. Elton, Thesis, Yale Univ. (1978).
Erik Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. MR 349393, DOI 10.2307/2272356
Fred Galvin and Karel Prikry, Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973), 193–198. MR 337630, DOI 10.2307/2272055
H. Knaust and E. Odell, On $c_0$ sequences in Banach spaces, Israel J. Math. 67 (1989), no. 2, 153–169. MR 1026560, DOI 10.1007/BF02937292
B. Maurey and H. P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977), no. 1, 77–98. MR 438091, DOI 10.4064/sm-61-1-77-98
C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33–39. MR 173640, DOI 10.1017/s0305004100038603
E. Odell, Applications of Ramsey theorems to Banach space theory, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 379–404. MR 606226
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
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 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
J. Schreier, Ein Gegenbeispiel zur theorie der schwachen konvergenz, Studia Math. 2 (1930), 58–62.
Jack Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60–64. MR 332480, DOI 10.2307/2271156
W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53–61. MR 227743, DOI 10.4064/sm-30-1-53-61
A. Tsarpalias, A note on the Ramsey property, Proc. Amer. Math. Soc. 127 (1999), no. 2, 583–587. MR 1458267, DOI 10.1090/S0002-9939-99-04518-9