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

• Autorzy: Elói Medina Galego , Maurício Zahn
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E40: Spaces of vector- and operator-valued functions
• 46B03: Isomorphic theory (including renorming) of Banach spaces
• 46B25: Classical Banach spaces in the general theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 141
• Numer: 1
• Strony: 51-59

Daty

wydano

2015

Twórcy

autor

Elói Medina Galego

• Department of Mathematics, IME, University of São Paulo, Rua do Matão, 1010, São Paulo, Brazil

autor

Maurício Zahn

• Department of Mathematics and Statistics, DME, Federal University of Pelotas, Campus Universitário Capão do Leão, s/n, Rio Grande do Sul, Brazil

Identyfikatory

DOI

10.4064/cm141-1-5