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

Abstract

It is shown that if $α,ζ$ are ordinals such that $1\leq ζ<α<ζω,$ 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 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.