A quasi-dichotomy for $C(\alpha , X)$ spaces, $\alpha < \omega _{1}$
Volume 141 / 2015
Colloquium Mathematicum 141 (2015), 51-59
MSC: Primary 46B03, 46E40; Secondary 46B25.
DOI: 10.4064/cm141-1-5
Abstract
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.
