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

Colloquium Mathematicae (2015)

• Volume: 141, Issue: 1, page 51-59
• ISSN: 0010-1354

top

## Abstract

top
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.

## How to cite

top

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 -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.