Some remarks on universality properties of $\ell_\infty / c_0$

Volume 128 / 2012

Mikołaj Krupski, Witold Marciszewski Colloquium Mathematicum 128 (2012), 187-195 MSC: Primary 46B26, 46E15; Secondary 03E75. DOI: 10.4064/cm128-2-4

Abstract

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 $\ell_\infty/c_0$. 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 $\ell_\infty/c_0$, but fails to embed isometrically. As far as we know it is the first example of this kind.

Authors

  • Mikołaj KrupskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    e-mail
  • Witold MarciszewskiInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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