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

Elói Medina Galego; Maurício Zahn

Colloquium Mathematicae (2015)

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

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topElói 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 -

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