Geometry of the Banach spaces $C(\beta {\mathbb N} \times K,X)$ for compact metric spaces $K$

Dale E. Alspach¹ ; Elói Medina Galego²

¹ Department of Mathematics Oklahoma State University Stillwater, OK 74078, U.S.A.
² Department of Mathematics University of São Paulo São Paulo, Brazil 05508-090

Studia Mathematica, Tome 207 (2011) no. 2, pp. 153-180

Résumé

A classical result of Cembranos and Freniche states that the $C(K, X)$ space contains a complemented copy of $c_{0}$ whenever $K$ is an infinite compact Hausdorff space and $X$ is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces $C(\alpha)$, $\alpha\omega_1$, are quotients of or complemented in $C(K,X)$.In contrast to the $c_0$ result, we prove that if $C(\beta \mathbb{N}\times [1,\omega], X)$ contains a complemented copy of $C(\omega^\omega)$ then $X$ contains a copy of $c_{0}$. Moreover, we show that $C(\omega^\omega)$ is not even a quotient of $C(\beta {\mathbb N} \times [1,\omega], \ell_p)$, $1 p \infty$.We then completely determine the separable $C(K)$ spaces which are isomorphic to a complemented subspace or a quotient of a $C(\beta {\mathbb N} \times [1,\alpha], \ell_p)$ space for countable ordinals $\alpha$ and $1 \leq p \infty$. As a consequence, we obtain the isomorphic classification of the $C(\beta {\mathbb N} \times K, \ell_p)$ spaces for infinite compact metric spaces $K$ and $1 \leq p \infty$. Indeed, we establish the following more general cancellation law. Suppose that the Banach space $X$ contains no copy of $c_{0}$ and $K_{1}$ and $K_{2}$ are infinite compact metric spaces, then the following statements are equivalent: (1) $C(\beta \mathbb{N}\times K_{1}, X)$ is isomorphic to $C(\beta \mathbb{N}\times K_{2}, X)$. (2) $C(K_{1})$ is isomorphic to $C(K_{2}).$ These results are applied to the isomorphic classification of some spaces of compact operators.

Détail

DOI : 10.4064/sm207-2-4

Keywords: classical result cembranos freniche states space contains complemented copy whenever infinite compact hausdorff space infinite dimensional banach space paper takes result starting point begins study conditions under which spaces alpha alpha omega quotients complemented contrast result prove beta mathbb times omega contains complemented copy omega omega contains copy moreover omega omega even quotient beta mathbb times omega ell infty completely determine separable spaces which isomorphic complemented subspace quotient beta mathbb times alpha ell space countable ordinals alpha leq infty consequence obtain isomorphic classification beta mathbb times ell spaces infinite compact metric spaces leq infty indeed establish following general cancellation law suppose banach space contains copy infinite compact metric spaces following statements equivalent beta mathbb times isomorphic beta mathbb times isomorphic these results applied isomorphic classification spaces compact operators