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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Universality of uniform Eberlein compacta
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by Mirna Džamonja PDF
Proc. Amer. Math. Soc. 134 (2006), 2427-2435 Request permission

Abstract:

We prove that if $\mu ^+ <\lambda =\textrm {cf}(\lambda )<\mu ^{\aleph _0}$ 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 Džamonja
  • Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
  • ORCID: setImmediate$0.3709267400444315$1
  • Email: h020@uea.ac.uk
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 2427-2435
  • MSC (2000): Primary 03E35, 03E75, 03C55, 54C35, 46E99
  • DOI: https://doi.org/10.1090/S0002-9939-06-08189-5
  • MathSciNet review: 2213717