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Separable quotients in $ C_{c}( X)$, $ C_{p}( X) $, and their duals

Authors: Jerzy Kakol and Stephen A. Saxon
Journal: Proc. Amer. Math. Soc. 145 (2017), 3829-3841
MSC (2010): Primary 46A08, 46A30, 54C35
Published electronically: May 24, 2017
MathSciNet review: 3665036
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Abstract: The quotient problem has a positive solution for the weak and strong duals of $ C_{c}\left ( X\right ) $ ($ X$ an infinite Tichonov space), for Banach spaces $ C_{c}\left ( X\right ) $, and even for barrelled $ C_{c}\left ( X\right ) $, but not for barrelled spaces in general. The solution is unknown for general $ C_{c}\left ( X\right ) $. A locally convex space is properly separable if it has a proper dense $ \aleph _{0}$-dimensional subspace. For $ C_{c}\left ( X\right ) $ quotients, properly separable coincides with infinite-dimensional separable. $ C_{c}\left ( X\right ) $ has a properly separable algebra quotient if $ X$ has a compact denumerable set. Relaxing compact to closed, we obtain the converse as well; likewise for $ C_{p}\left ( X\right ) $. And the weak dual of $ C_{p}\left ( X\right ) $, which always has an $ \aleph _{0}$-dimensional quotient, has no properly separable quotient when $ X$ is a P-space of a certain special form $ X=X_\kappa $

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Additional Information

Jerzy Kakol
Affiliation: Faculty of Mathematics and Informatics, A. Mickiewicz University, 60-769 Poznań, Matejki 48-49, Poland – and Institute of Mathematics, Czech Academy of Sciences, Zitna 25, Prague, Czech Republic

Stephen A. Saxon
Affiliation: Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105

Keywords: (Properly) separables quotients, $C( X) $, weak barrelledness, P-spaces
Received by editor(s): May 3, 2016
Received by editor(s) in revised form: June 1, 2016, and June 21, 2016
Published electronically: May 24, 2017
Additional Notes: Thanks to Professor Aaron R. Todd for vital discussions/encouragement/prequels.
The first author’s research was supported by Generalitat Valenciana, Conselleria d’Educació, Cultura i Esport, Spain, grant PROMETEO/2013/058, and by GACR grant 16-34860L and RVO 67985840 (Czech Republic).
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society