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Projections in duals to Asplund spaces made without Simons' lemma

Authors: Marek Cúth and Marián Fabian
Journal: Proc. Amer. Math. Soc. 143 (2015), 301-308
MSC (2010): Primary 46B26; Secondary 46B20, 46B22
Published electronically: September 12, 2014
MathSciNet review: 3272755
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Abstract: G. Godefroy and the second author of this note proved in 1988 that in duals to Asplund spaces there always exists a projectional resolution of the identity. A few years later, Ch. Stegall succeeded to omit from the original proof a deep lemma of S. Simons. Here, we rewrite the condensed argument of Ch. Stegall in a more transparent and detailed way. We actually show that this technology of Ch. Stegall leads to a bit stronger/richer object--the so-called projectional skeleton--recently constructed by W. Kubiś, via S. Simons' lemma and with the help of elementary submodels from logic.

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Additional Information

Marek Cúth
Affiliation: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 18675 Praha 8, Czech Republic

Marián Fabian
Affiliation: Mathematical Institute of Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

Keywords: Asplund space, projectional resolution of identity, projectional skeleton, weak star dentability, Jayne-Rogers selection theorem, separable reduction
Received by editor(s): April 4, 2013
Published electronically: September 12, 2014
Additional Notes: The first author was supported by Grant No. 282511/B-MAT/MFF of the Grant Agency of Charles University in Prague.
The second author was supported by grant P201/12/0290 and by RVO: 67985840
Dedicated: Dedicated to the 70th birthday of Charles Stegall
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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