## Universality of uniform Eberlein compacta

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## 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.## References

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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