A quasi-dichotomy for C(α,X) spaces, α < ω₁

Elói Medina Galego, and Maurício Zahn. "A quasi-dichotomy for C(α,X) spaces, α < ω₁." Colloquium Mathematicae 141.1 (2015): 51-59. <http://eudml.org/doc/286399>.

@article{ElóiMedinaGalego2015,
abstract = { 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. },
author = {Elói Medina Galego, Maurício Zahn},
journal = {Colloquium Mathematicae},
keywords = {separable spaces; spaces of finite cotype},
language = {eng},
number = {1},
pages = {51-59},
title = {A quasi-dichotomy for C(α,X) spaces, α < ω₁},
url = {http://eudml.org/doc/286399},
volume = {141},
year = {2015},
}

TY - JOUR
AU - Elói Medina Galego
AU - Maurício Zahn
TI - A quasi-dichotomy for C(α,X) spaces, α < ω₁
JO - Colloquium Mathematicae
PY - 2015
VL - 141
IS - 1
SP - 51
EP - 59
AB - 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.
LA - eng
KW - separable spaces; spaces of finite cotype
UR - http://eudml.org/doc/286399
ER -