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On the topology of pointwise convergence on the boundaries of $ L_1$-preduals


Authors: Warren B. Moors and Jirí Spurny
Journal: Proc. Amer. Math. Soc. 137 (2009), 1421-1429
MSC (2000): Primary 46A50; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-08-09708-6
Published electronically: October 29, 2008
MathSciNet review: 2465668
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove a theorem more general than the following:

``If $ (X,\Vert\cdot\Vert)$ is an $ L_1$-predual, $ B$ is any boundary of $ X$ and $ \{x_n:n \in \N\}$ is any subset of $ X$, then the closure of $ \{x_n:n \in \N\}$ with respect to the topology of pointwise convergence on $ B$ is separable with respect to the topology generated by the norm, whenever $ {\rm Ext}(B_{X^*})$ is weak$ ^*$ Lindelöf.'' Several applications of this result are also presented.


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  • 1. Erik M. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57, Springer-Verlag, New York-Heidelberg, 1971. MR 0445271 (56:3615)
  • 2. Jonathan M. Borwein and Warren B. Moors, Separable determination of integrability and minimality of the Clarke subdifferential mapping, Proc. Amer. Math. Soc. 128 (2000), 215-221. MR 1622793 (2000c:49025)
  • 3. Bernardo Cascales and Gilles Godefroy, Angelicity and the boundary problem, Mathematika 45 (1998), 105-112. MR 1644346 (99f:46019)
  • 4. V. P. Fonf, J. Lindenstrauss and R. R. Phelps, Infinite dimensional convexity, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, 599-670. MR 1863703 (2003c:46014)
  • 5. S. S. Khurana, Pointwise compactness on extreme points, Proc. Amer. Math. Soc. 83 (1981), 347-348. MR 624928 (82k:46018)
  • 6. H. E. Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der Mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279 (58:12308)
  • 7. Peijie Lin and Warren B. Moors, Rich families and the product of Baire spaces, Math. Balkanica, to appear, available at http://www.math.auckland.ac.nz/$ \sim$moors/.
  • 8. Warren B. Moors, A characterisation of weak compactness in Banach spaces, Bull. Austral. Math. Soc. 55 (1997), 497-501. MR 1456278 (98g:46019)
  • 9. Warren B. Moors and Evgenii A. Reznichenko, Separable subspaces of affine function spaces on compact convex sets, Topology Appl. 155 (2008), 1306-1322.
  • 10. R. Phelps, Lectures on Choquet's Theorem, second edition, Lecture Notes in Mathematics, no. 1757, Springer-Verlag, Berlin, 2001. MR 1835574 (2002k:46001)
  • 11. Evgenii A. Reznichenko, Compact convex spaces and their maps, Topology Appl. 36 (1990), 117-141. MR 1068165 (91g:54010)
  • 12. J. Saint Raymond, Jeux topologiques et espaces de Namioka, Proc. Amer. Math. Soc. 87 (1983), 449-504. MR 684646 (83m:54060)
  • 13. Jiří Spurný, The boundary problem for $ L_1$-preduals. Illinois J. Math. to appear in 2009, available at http://www.karlin.mff.cuni.cz/kma-preprints/.

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Additional Information

Warren B. Moors
Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: moors@math.auckland.ac.nz

Jirí Spurny
Affiliation: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: spurny@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/S0002-9939-08-09708-6
Keywords: Compact convex, extreme points, boundary, $L_1$-predual.
Received by editor(s): May 22, 2008
Published electronically: October 29, 2008
Additional Notes: The second author was supported by the research project MSM 0021620839 financed by MSMT and by the grant GAČR 201/07/0388.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society

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