We prove the following quasi-dichotomy involving the Banach spaces C(α,X) of all X-valued continuous functions defined on the interval [0,α] of ordinals and endowed with the supremum norm. Suppose that X and Y are arbitrary Banach spaces of finite cotype. Then at least one of the following statements is true. (1) There exists a finite ordinal n such that either C(n,X) contains a copy of Y, or C(n,Y) contains a copy of X. (2) For any infinite countable ordinals α, β, ξ, η, the following are equivalent: (a) C(α,X) ⊕ C(ξ,Y) is isomorphic to C(β,X) ⊕ C(η,Y). (b) C(α) is isomorphic to C(β), and C(ξ) is isomorphic to C(η). This result is optimal in the sense that it cannot be extended to uncountable ordinals.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-1-5,
author = {El\'oi Medina Galego and Maur\'\i cio Zahn},
title = {A quasi-dichotomy for C(a,X) spaces, a < o1},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {51-59},
zbl = {1342.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-1-5}
}
Elói Medina Galego; Maurício Zahn. A quasi-dichotomy for C(α,X) spaces, α < ω₁. Colloquium Mathematicae, Tome 139 (2015) pp. 51-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-1-5/