A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ 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(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of then X contains a copy of c₀. Moreover, we show that is not even a quotient of , 1 < p < ∞. We then completely determine the separable C(K) spaces which are isomorphic to a complemented subspace or a quotient of a space for countable ordinals α and 1 ≤ p < ∞. As a consequence, we obtain the isomorphic classification of the spaces for infinite compact metric spaces K and 1 ≤ p < ∞. Indeed, we establish the following more general cancellation law. Suppose that the Banach space X contains no copy of c₀ and K₁ and K₂ are infinite compact metric spaces, then the following statements are equivalent: (1) C(βℕ ×K₁,X) is isomorphic to C(βℕ ×K₂,X). (2) C(K₁) is isomorphic to C(K₂). These results are applied to the isomorphic classification of some spaces of compact operators.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm207-2-4,
author = {Dale E. Alspach and El\'oi Medina Galego},
title = {Geometry of the Banach spaces C($\beta$$\mathbb{N}$ $\times$ K,X) for compact metric spaces K},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {153-180},
zbl = {1246.46009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm207-2-4}
}
Dale E. Alspach; Elói Medina Galego. Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K. Studia Mathematica, Tome 204 (2011) pp. 153-180. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm207-2-4/