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Banach spaces with a unique nontrivial decomposition


Authors: Spiros A. Argyros and Theocharis Raikoftsalis
Journal: Proc. Amer. Math. Soc. 136 (2008), 3611-3620
MSC (2000): Primary 46B20, 46B26
DOI: https://doi.org/10.1090/S0002-9939-08-09368-4
Published electronically: June 2, 2008
MathSciNet review: 2415045
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Abstract: Motivated by a problem of P. Koszmider we introduce the class of quasi-prime Banach spaces. This class lies between the classes of prime and primary Banach spaces. It is shown that for every $ 1<p<\infty$ there exists a strictly quasi-prime separable reflexive Banach space $ \mathfrak{X}_p$ such that $ \ell_p$ is a complemented subspace of $ \mathfrak{X}_p$. A similar result also holds for the case of $ \ell_1$ and $ c_0$. More generally, for every separable decomposable prime Banach space $ Y$ not containing $ \ell_1$ there exists a strictly quasi-prime $ \mathfrak{X}_Y$ containing $ Y$ as a complemented subspace. We also investigate the operators acting on these spaces as well as the complemented subspaces of their finite powers.


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  • 1. D. Alspach and S.A. Argyros, Complexity of weakly null sequences, Dissertationes Mathematicae, 321(1992), 1-44. MR 1191024 (93j:46014)
  • 2. S.A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, Journal AMS, 13(2001), 243-294. MR 1750954 (2002b:46021)
  • 3. S.A. Argyros, S. Merkourakis and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Isr. J. Math., 107(1998), 157-193. MR 1658551 (99m:46021)
  • 4. S.A. Argyros and S. Todorcevic, Ramsey methods in analysis, Birkhäuser, CRM, Barcelona, 2004. MR 2145246 (2006e:46015)
  • 5. S.A. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Memoirs of the AMS, 170(2004), no. 806. MR 2053392 (2005f:46022)
  • 6. W.T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc., 6(1993), no. 4, 851-874. MR 1201238 (94k:46021)
  • 7. W.T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann., 307(1997), no. 4, 543-568. MR 1464131 (98g:46018)
  • 8. P. Koszmider, On decompositions of Banach spaces of continuous functions on Mrowka's spaces, Proc. AMS, 133(2005), 2137-2146. MR 2137881 (2006a:46030)
  • 9. J. Lindenstrauss, On complemented subspaces of $ m$, Israel J. Math., 5(1967), 153-156. MR 0222616 (36:5666)
  • 10. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I and II, Springer, 1996. MR 0500056 (58:17766); MR 0540367 (81c:46001)
  • 11. B. Maurey, Banach spaces with few operators, Handbook of the geometry of Banach spaces, Vol. 2, 1247-1297, North-Holland, Amsterdam, 2003. MR 1999196 (2004m:46014)
  • 12. A. Pelczynski, Projections in certain Banach spaces, Studia Math., 9(1960), 209-228. MR 0126145 (23:A3441)
  • 13. Th. Schlumprecht, An arbitrarily distortable space, Israel J. Math., 76(1991), 81-95. MR 1177333 (93h:46023)

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Additional Information

Spiros A. Argyros
Affiliation: Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, National Technical University of Athens, 157 80, Athens, Greece
Email: sargyros@math.ntua.gr

Theocharis Raikoftsalis
Affiliation: Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, National Technical University of Athens, 157 80, Athens, Greece
Email: th-raik@hotmail.com

DOI: https://doi.org/10.1090/S0002-9939-08-09368-4
Keywords: Prime Banach space, primary Banach space, Schauder decomposition, hereditarily indecomposable Banach space, strictly singular operators, complemented subspaces
Received by editor(s): July 12, 2007
Received by editor(s) in revised form: September 11, 2007
Published electronically: June 2, 2008
Additional Notes: This work was partially supported by the Leukippos NTUA Research programme.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society

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