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

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 $\ell_\infty/c_0$.