Universality of uniform Eberlein compacta

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.