## Remarks about Schlumprecht space

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- by Denka Kutzarova and Pei-Kee Lin
- Proc. Amer. Math. Soc.
**128**(2000), 2059-2068 - DOI: https://doi.org/10.1090/S0002-9939-99-05248-X
- Published electronically: December 8, 1999
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## 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.

## Bibliographic Information

**Denka Kutzarova**- Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
- Address at time of publication: Department of Mathematics, University of South Carolina, Columbia, South Carolina
- MR Author ID: 108570
- Email: denka@math.sc.edu
**Pei-Kee Lin**- Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152
- Email: linpk@msci.memphis.edu
- Received by editor(s): August 3, 1998
- Received by editor(s) in revised form: August 27, 1998
- Published electronically: December 8, 1999
- Additional Notes: Part of this paper was done when the second author visited the University of Texas at Austin and was completed when the first author participated in the Workshop in Linear Analysis and Probability Theory at Texas A&M University, 1998. Both authors would like to thank E. Odell and Th. Schlumprecht for their valuable discussions
- Communicated by: Dale Alspach
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**128**(2000), 2059-2068 - MSC (2000): Primary 46B20, 46B45
- DOI: https://doi.org/10.1090/S0002-9939-99-05248-X
- MathSciNet review: 1654081