Some remarks on universality properties of $\ell_\infty / c_0$

TL;DR

The paper investigates the universality properties of the Banach space \\ell__0\\, demonstrating that under certain set-theoretic assumptions, there exist compact spaces whose continuous function spaces do not embed into \\ell__0\\, either isometrically or isomorphically.

Contribution

It establishes new results linking set theory and the embedding properties of Banach spaces of continuous functions on compacta, including the first known example of a compactum with specific embedding characteristics.

Findings

Existence of a uniform Eberlein compact space with no isometric embedding into \\ell__0\\.

Existence of a uniform Eberlein compact space whose continuous functions embed isomorphically but not isometrically.

First known example of a compactum with such embedding properties.

Abstract

We prove that if continuum is not a Kunen cardinal, then there is a uniform Eberlein compact space $K$ such that the Banach space $C(K)$ does not embed isometrically into $\ell_\infty/c_0$. We prove a similar result for isomorphic embeddings. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $\ell_\infty/c_0$, but fails to embed isometrically. As far as we know it is the first example of this kind.