Universality of uniform Eberlein compacta

Abstract

We prove that if $\mu^+ 2^{\aleph_0}$, 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^{++}, 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