$\ell_{\infty} $-sums and the Banach space $\ell_{\infty}/ c_{0}$
Tom 224 / 2014
Streszczenie
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 with 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 $\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 $\ell _\infty /c_0$.
