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Isomorphisms of spaces of continuous affine functions on compact convex sets with Lindelöf boundaries


Authors: Pavel Ludvík and Jiří Spurný
Journal: Proc. Amer. Math. Soc. 139 (2011), 1099-1104
MSC (2010): Primary 46A55, 46E15, 54D20
DOI: https://doi.org/10.1090/S0002-9939-2010-10534-8
Published electronically: August 10, 2010
MathSciNet review: 2745661
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Abstract: Let $ X,Y$ be compact convex sets such that every extreme point of $ X$ and $ Y$ is a weak peak point and both $ \operatorname{ext} X$ and $ \operatorname{ext} Y$ are Lindelöf spaces. We prove that if there exists an isomorphism $ T:\mathfrak{A}^c(X)\to \mathfrak{A}^c(Y)$ with $ \Vert T\Vert\cdot \Vert T^{-1}\Vert<2$, then $ \operatorname{ext} X$ is homeomorphic to $ \operatorname{ext} Y$. This generalizes results of C. H. Chu and H. B. Cohen.


References [Enhancements On Off] (What's this?)

  • 1. E. M. Alfsen.
    Compact convex sets and boundary integrals.
    Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57,
    Springer-Verlag, New York, 1971. MR 0445271 (56:3615)
  • 2. L. Asimow and A. J. Ellis.
    Convexity theory and its applications in functional analysis, volume 16 of London Mathematical Society Monographs.
    Academic Press, Inc. [Harcourt Brace Jovanovich Publishers], London, 1980. MR 623459 (82m:46009)
  • 3. E. Bishop and K. de Leeuw.
    The representations of linear functionals by measures on sets of extreme points.
    Ann. Inst. Fourier (Grenoble), 9:305-331, 1959. MR 0114118 (22:4945)
  • 4. G. Choquet.
    Lectures on analysis. Vols. I-III: Infinite dimensional measures and problem solutions.
    Edited by J. Marsden, T. Lance and S. Gelbart. W. A. Benjamin, Inc., New York-Amsterdam, 1969.
  • 5. C. H. Chu and H. B. Cohen.
    Isomorphisms of spaces of continuous affine functions.
    Pacific J. Math., 155(1):71-85, 1992. MR 1174476 (93i:46041)
  • 6. D. H. Fremlin.
    Measure theory. Vol. 4.
    Torres Fremlin, Colchester, 2006.
    Topological measure spaces. Parts I, II. Corrected second printing of the 2003 original. MR 2462372
  • 7. H. U. Hess.
    On a theorem of Cambern.
    Proc. Amer. Math. Soc., 71(2):204-206, 1978. MR 500490 (80e:46010)
  • 8. U. Krause.
    Der Satz von Choquet als ein abstrakter Spektralsatz und vice versa.
    Math. Ann., 184(4):275-296, 1970. MR 1513280
  • 9. John C. Oxtoby.
    Measure and category. A survey of the analogies between topological and measure spaces.
    Graduate Texts in Mathematics, Vol. 2,
    Springer-Verlag, New York, 1971. MR 0393403 (52:14213)
  • 10. J. Spurný.
    Baire classes of Banach spaces and strongly affine functions.
    Trans. Amer. Math. Soc., 362(3):1659-1680, 2010. MR 2563744

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Additional Information

Pavel Ludvík
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: ludvik@karlin.mff.cuni.cz

Jiří Spurný
Affiliation: Department of Mathematical Analysis, 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/S0002-9939-2010-10534-8
Keywords: Compact convex set, extreme point, weak peak point, Lindelöf space, continuous affine function
Received by editor(s): January 7, 2010
Received by editor(s) in revised form: April 9, 2010
Published electronically: August 10, 2010
Additional Notes: The first author was supported by grant GAČR 401/09/H007.
The second author was supported in part by the grants GAAV IAA 100190901 and GAČR 201/07/0388, and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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