A Banach space with the Schur and the Daugavet property
HTML articles powered by AMS MathViewer
- by Vladimir Kadets and Dirk Werner PDF
- Proc. Amer. Math. Soc. 132 (2004), 1765-1773 Request permission
Abstract:
We show that a minor refinement of the Bourgain-Rosenthal construction of a Banach space without the Radon-Nikodým property which contains no bounded $\delta$-trees yields a space with the Daugavet property and the Schur property. Using this example we answer some open questions on the structure of such spaces; in particular, we show that the Daugavet property is not inherited by ultraproducts.References
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
- D. Bilik, V. M. Kadets, R. V. Shvidkoy and D. Werner. Narrow operators and the Daugavet property for ultraproducts. To appear in Positivity. Preprint available from http://xxx.lanl.gov.
- J. Bourgain and H. P. Rosenthal, Martingales valued in certain subspaces of $L^{1}$, Israel J. Math. 37 (1980), no. 1-2, 54–75. MR 599302, DOI 10.1007/BF02762868
- Vladimir M. Kadets, Some remarks concerning the Daugavet equation, Quaestiones Math. 19 (1996), no. 1-2, 225–235. MR 1390483
- V. M. Kadets, N. Kalton, and D. Werner. Remarks on rich subspaces of Banach spaces. To appear in Studia Math. Preprint available from http://xxx.lanl.gov.
- Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin, and Dirk Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), no. 2, 855–873. MR 1621757, DOI 10.1090/S0002-9947-99-02377-6
- Vladimir M. Kadets, Roman V. Shvidkoy, and Dirk Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math. 147 (2001), no. 3, 269–298. MR 1853772, DOI 10.4064/sm147-3-5
- Klaus D. Schmidt, Daugavet’s equation and orthomorphisms, Proc. Amer. Math. Soc. 108 (1990), no. 4, 905–911. MR 1002165, DOI 10.1090/S0002-9939-1990-1002165-3
- R. V. Shvydkoy, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), no. 2, 198–212. MR 1784413, DOI 10.1006/jfan.2000.3626
- M. Gontcharoff, Sur quelques séries d’interpolation généralisant celles de Newton et de Stirling, Uchenye Zapiski Moskov. Gos. Univ. Matematika 30 (1939), 17–48 (Russian, with French summary). MR 0002002
- P. Wojtaszczyk, Some remarks on the Daugavet equation, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1047–1052. MR 1126202, DOI 10.1090/S0002-9939-1992-1126202-2
Additional Information
- Vladimir Kadets
- Affiliation: Faculty of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine
- Address at time of publication: Department of Mathematics, Freie Universität Berlin, Arnimallee 2–6, D-14 195 Berlin, Germany
- MR Author ID: 202226
- ORCID: 0000-0002-5606-2679
- Email: vova1kadets@yahoo.com, kadets@math.fu-berlin.de
- Dirk Werner
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 2–6, D-14 195 Berlin, Germany
- Email: werner@math.fu-berlin.de
- Received by editor(s): February 13, 2003
- Published electronically: October 24, 2003
- Additional Notes: The work of the first author was supported by a fellowship from the Alexander-von-Humboldt Stiftung.
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1765-1773
- MSC (2000): Primary 46B04; Secondary 46B20, 46M07
- DOI: https://doi.org/10.1090/S0002-9939-03-07278-2
- MathSciNet review: 2051139