Some remarks on universality properties of ${\ell }_{\mathrm{\infty }}/c_0$

Mikołaj Krupski¹ ; Witold Marciszewski²

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland
² Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 187-195.

Résumé

We prove that if $\mathfrak{c}$ 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 ${\ell }_{\mathrm{\infty }}/c_0$. We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into ${\ell }_{\mathrm{\infty }}/c_0$, but fails to embed isometrically. As far as we know it is the first example of this kind.

Détail

DOI : 10.4064/cm128-2-4

Mots-clés : prove mathfrak kunen cardinal there uniform eberlein compact space banach space does embed isometrically ell infty prove similar result isomorphic embeddings arguments minor modifications proofs analogous results corson compacta obtained todor evi construct consistent example uniform eberlein compactum whose space continuous functions embeds isomorphically ell infty fails embed isometrically far know first example kind