Remarks about Schlumprecht space

Kutzarova, Denka; Lin, Pei-Kee

doi:10.1090/S0002-9939-99-05248-X

Remarks about Schlumprecht space
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by Denka Kutzarova and Pei-Kee Lin PDF

Proc. Amer. Math. Soc. 128 (2000), 2059-2068 Request permission

Abstract:

Let $\mathbf S$ denote the Schlumprecht space. We prove that (1) $\ell _\infty$ is finitely disjointly representable in $\mathbf S$; (2) $\mathbf S$ contains an $\ell _1$-spreading model; (3) for any sequence $(n_k)$ of natural numbers, $\mathbf S$ is isomorphic to the space $(\sum _{k=1}^\infty \oplus \ell _\infty ^{n_k})_{\mathbf S}$.

References

S. A. Argyros, I. Deliyanni, D. N. Kutzarova and A. Manoussakis, Modified mixed Tsirelson spaces, J. Funct. Anal. 159 (1998), 43–109.
B. Beauzamy and J.-T. Lapresté, Modèles étalés des espaces de Banach, Travaux en Cours. [Works in Progress], Hermann, Paris, 1984 (French). MR 770062
G. Bennett, L. E. Dor, V. Goodman, W. B. Johnson, and C. M. Newman, On uncomplemented subspaces of $L_{p},$ $1<p<2$, Israel J. Math. 26 (1977), no. 2, 178–187. MR 435822, DOI 10.1007/BF03007667
J. Bourgain, A counterexample to a complementation problem, Compositio Math. 43 (1981), no. 1, 133–144. MR 631431
J. Bourgain, P. G. Casazza, J. Lindenstrauss, and L. Tzafriri, Banach spaces with a unique unconditional basis, up to permutation, Mem. Amer. Math. Soc. 54 (1985), no. 322, iv+111. MR 782647, DOI 10.1090/memo/0322
Peter G. Casazza, The Schroeder-Bernstein property for Banach spaces, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 61–77. MR 983381, DOI 10.1090/conm/085/983381
P. G. Casazza and N. J. Kalton, Uniqueness of unconditional bases in Banach spaces, Israel J. Math. 103 (1998), 141–175. MR 1613564, DOI 10.1007/BF02762272
W. T. Gowers, A solution to Banach’s hyperplane problem, Bull. London Math. Soc. 26 (1994), no. 6, 523–530. MR 1315601, DOI 10.1112/blms/26.6.523
W. T. Gowers, A solution to the Schroeder-Bernstein problem for Banach spaces, Bull. London Math. Soc. 28 (1996), no. 3, 297–304. MR 1374409, DOI 10.1112/blms/28.3.297
W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
W. T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997), no. 4, 543–568. MR 1464131, DOI 10.1007/s002080050050
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
E. Odell and Th. Schlumprecht, On the richness of the set of $p$’s in Krivine’s theorem, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 177–198. MR 1353459
Haskell P. Rosenthal, On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303. MR 271721, DOI 10.1007/BF02771562
Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81–95. MR 1177333, DOI 10.1007/BF02782845
Th. Schlumprecht, A complementably minimal Banach space not containing $c_0$ or $\ell _p$, Seminar notes Louisiana State University, 1991/2, 169–181.