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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A weak$ ^*$ separable $ C(K)^*$ space whose unit ball is not weak$ ^*$ separable


Authors: A. Avilés, G. Plebanek and J. Rodríguez
Journal: Trans. Amer. Math. Soc. 366 (2014), 4733-4753
MSC (2010): Primary 28E15, 46E15, 46E27, 54G20
Published electronically: May 5, 2014
MathSciNet review: 3217698
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Abstract: We provide a ZFC example of a compact space $ K$ such that $ C(K)^*$ is $ w^*$-separable but its closed unit ball $ B_{C(K)^*}$ is not $ w^*$-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that $ C(K)$ equipped with its weak Baire $ \sigma $-algebra.


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Additional Information

A. Avilés
Affiliation: Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: avileslo@um.es

G. Plebanek
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland
Email: grzes@math.uni.wroc.pl

J. Rodríguez
Affiliation: Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: joserr@um.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05962-X
PII: S 0002-9947(2014)05962-X
Keywords: Weak$^*$-separability, Baire $\sigma$-algebra, measurability, Kunen cardinal
Received by editor(s): December 23, 2011
Received by editor(s) in revised form: September 19, 2012
Published electronically: May 5, 2014
Additional Notes: The first and third authors were supported by MEC and FEDER (Project MTM2008-05396) and Fundación Séneca (Project 08848/PI/08). The first author was supported by Ramon y Cajal contract (RYC-2008-02051) and an FP7-PEOPLE-ERG-2008 action. The second author was supported by MNiSW Grant N N201 418939 (2010–2013)
Article copyright: © Copyright 2014 American Mathematical Society