Geometry of the Banach spaces $C(\beta {\mathbb N} \times K,X)$ for compact metric spaces $K$
Volume 207 / 2011
Abstract
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.
