$\ell_\infty$-sums and the Banach space $\ell_\infty/c_0$

TL;DR

This paper investigates the complex structure of the Banach space /c_0, showing that its decomposition properties depend on set-theoretic assumptions beyond ZFC, and demonstrating the existence of models where certain embeddings do not exist.

Contribution

It establishes the consistency of /c_0 lacking an orthogonal -decomposition and of (c_0(\u03c4)) not embedding into /c_0, revealing set-theoretic dependence of its structure.

Findings

/c_0 may lack an -decomposition under certain set-theoretic assumptions.

(c_0()) does not necessarily embed into /c_0 in some models.

Under continuum hypothesis, /c_0 contains all (X) spaces for subspaces X.

Abstract

This paper is concerned with the isomorphic structure of the Banach space $\ell_\infty/c_0$ and how it depends on combinatorial tools whose existence is consistent but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $\ell_\infty/c_0$ does not have an orthogonal \break $\ell_\infty$-decomposition that is, it is not of the form $\ell_\infty(X)$ for any Banach space $X$. The main local result is that it is consistent that $\ell_\infty(c_0(\mathfrak{c}))$ does not embed isomorphically into $\ell_\infty/c_0$, where $\mathfrak{c}$ is the cardinality of the continuum, while $\ell_\infty$ and $c_0(\mathfrak{c})$ always do embed quite canonically. This should be compared with the results of Drewnowski and Roberts that under the assumption of the continuum hypothesis $\ell_\infty/c_0$ is isomorphic to its $\ell_\infty$-sum and in particular it contains an isomorphic copy of all Banach spaces of the form $\ell_\infty(X)$ for any subspace $X$ of $\ell_\infty/c_0$.