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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Universality of uniform Eberlein compacta


Author: Mirna Dzamonja
Journal: Proc. Amer. Math. Soc. 134 (2006), 2427-2435
MSC (2000): Primary 03E35, 03E75, 03C55, 54C35, 46E99
Published electronically: January 31, 2006
MathSciNet review: 2213717
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Abstract: We prove that if $ \mu^+ <\lambda={\rm cf}(\lambda)<\mu^{\aleph_0}$ for some regular $ \mu>2^{\aleph_0}$, then there is no family of less than $ \mu^{\aleph_0}$ c-algebras of size $ \lambda$ which are jointly universal for c-algebras of size $ \lambda$. On the other hand, it is consistent to have a cardinal $ \lambda\ge \aleph_1$ as large as desired and satisfying $ \lambda^{<\lambda}=\lambda$ and $ 2^{\lambda^+}>\lambda^{++}$, while there are $ \lambda^{++}$ c-algebras of size $ \lambda^+$ that are jointly universal for c-algebras of size $ \lambda^+$. Consequently, by the known results of M. Bell, it is consistent that there is $ \lambda$ as in the last statement and $ \lambda^{++}$ uniform Eberlein compacta of weight $ \lambda^+$ such that at least one among them maps onto any Eberlein compact of weight $ \lambda^+$ (we call such a family universal). The only positive universality results for Eberlein compacta known previously required the relevant instance of $ GCH$ to hold. These results complete the answer to a question of Y. Benyamini, M. E. Rudin and M. Wage from 1977 who asked if there always was a universal uniform Eberlein compact of a given weight.


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Additional Information

Mirna Dzamonja
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
Email: h020@uea.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08189-5
PII: S 0002-9939(06)08189-5
Keywords: Universal models, c-algebras, Eberlein compacta.
Received by editor(s): February 27, 2002
Received by editor(s) in revised form: February 23, 2005
Published electronically: January 31, 2006
Additional Notes: The author thanks EPSRC for their support through the grant number GR/M71121 and the EPSRC Advanced Fellowship, and the referees for their comments on the paper.
Communicated by: Alan Dow
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.