$\ell_{\infty} $-sums and the Banach space $\ell_{\infty}/ c_{0}$

Christina Brech¹ ; Piotr Koszmider²

¹ Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo Caixa Postal 66281 05314-970, São Paulo, Brazil
² Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 224 (2014) no. 2, pp. 175-185.

Résumé

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 $\ell _\infty /c_0$.

Détail

DOI : 10.4064/fm224-2-3

Mots-clés : paper concerned isomorphic structure banach space ell infty depends combinatorial tools whose existence consistent provable usual axioms zfc main global result consistent ell infty does have orthogonal ell infty decomposition form ell infty banach space main local result consistent ell infty mathfrak does embed isomorphically ell infty where mathfrak cardinality continuum while ell infty mathfrak always embed quite canonically should compared results drewnowski roberts under assumption continuum hypothesis ell infty isomorphic its ell infty sum particular contains isomorphic copy banach spaces form ell infty subspace ell infty