Operators on $C({\omega }^{\alpha })$ which do not preserve $C({\omega }^{\alpha })$

Abstract

It is shown that if $\alpha ,\zeta$ are ordinals such that $1\le \zeta <\alpha <\zeta \omega ,$ then there is an operator from $C({\omega }^{{\omega }^{\alpha }})$ onto itself such that if $Y$ is a subspace of $C({\omega }^{{\omega }^{\alpha }})$ which is isomorphic to $C({\omega }^{{\omega }^{\alpha }})$ $,$ 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 $\alpha$ for which any operator from $C({\omega }^{{\omega }^{\alpha }})$ onto itself there is a subspace of $C({\omega }^{{\omega }^{\alpha }})$ which is isomorphic to on which the operator is an isomorphism.