A quasi-dichotomy for $C(\alpha , X)$ spaces, $\alpha \omega _{1}$

Elói Medina Galego ¹   ; Maurício Zahn ²

¹ Department of Mathematics, IME University of São Paulo Rua do Matão 1010, São Paulo, Brazil
² Department of Mathematics and Statistics, DME Federal University of Pelotas Campus Universitário Capão do Leão, s/n Rio Grande do Sul, Brazil

Colloquium Mathematicum, Tome 141 (2015) no. 1, pp. 51-59

Résumé

We prove the following quasi-dichotomy involving the Banach spaces $C(\alpha, X)$ of all $X$-valued continuous functions defined on the interval $[0, \alpha]$ 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 $\alpha$, $\beta$, $\xi$, $\eta$, the following are equivalent: (a) $C(\alpha, X) \oplus C(\xi, Y)$ is isomorphic to $C(\beta, X) \oplus C(\eta, Y)$. (b) $C(\alpha)$ is isomorphic to $C(\beta)$, and $C(\xi)$ is isomorphic to $C(\eta).$This result is optimal in the sense that it cannot be extended to uncountable ordinals.

Détail

DOI : 10.4064/cm141-1-5

Keywords: prove following quasi dichotomy involving banach spaces alpha x valued continuous functions defined interval alpha ordinals endowed supremum norm suppose arbitrary banach spaces finite cotype least following statements there exists finite ordinal either contains copy nbsp contains copy nbsp infinite countable ordinals alpha beta eta following equivalent alpha oplus isomorphic beta oplus eta alpha isomorphic beta isomorphic eta result optimal sense cannot extended uncountable ordinals