Operators on $C(ω^α)$ which do not preserve $C(ω^α)$

Dale E. Alspach¹

¹

Fundamenta Mathematicae, Tome 153 (1997) no. 1, pp. 81-98.

Résumé

It is shown that if α,ζ are ordinals such that 1 ≤ ζ α ζω, then there is an operator from $C(ω^{ω^α})$ onto itself such that if Y is a subspace of $C(ω^{ω^α})$ which is isomorphic to $C(ω^{ω^α})$, then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C(ω^{ω^α})$ onto itself there is a subspace of $C(ω^{ω^α})$ which is isomorphic to $C(ω^{ω^α})$ on which the operator is an isomorphism.

Détail

DOI : 10.4064/fm-153-1-81-98

Keywords: ordinal index, Szlenk index, Banach space of continuous functions