Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, the Banach ultrapower $C(K)_\mathcal{U}$ is isomorphic to $\ell_\infty/c_0$ for many metrizable $K$. I wonder if it is possible to get this result for some $\mathcal{U}$ without assuming CH.

More precisely, given an infinite, metrizable, totally disconnected compact space $K$ having no isolated points, and a nontrivial ultrafilter $\mathcal{U}$ on $\mathbb{N}$, the ultracoproduct $K^\mathcal{U}$ is a Parovicenko space. Under CH, the only Parovicenko space is $\beta N\setminus N$, so $$ C(K)_\mathcal{U} \equiv C(K^\mathcal{U}) \equiv C(\beta N\setminus N) \equiv\ell_\infty/c_0. $$ Without CH, there are many Parovicenko spaces. I want to know if it is posible to show without CH that $K^\mathcal{U}=\beta N\setminus N$, or that $C(K^\mathcal{U})$ is isomorphic to $\ell_\infty/c_0$, for some $\mathcal{U}$.

isometricto $\ell_\infty / c_0$. As for the isomorphic case, it seems that there are also $2^{\mathfrak{c}}$ isomorphic types by a version of Theorem 3 from arxiv.org/pdf/0912.0406.pdf adjusted to the setting of ptmat.fc.ul.pt/~alexus/papers/unstable.pdf – Tomek Kania Sep 14 at 20:02