Alspach, Dale E. Operators on \(C(\omega^\alpha)\) which do not preserve \(C(\omega^\alpha)\). (English) Zbl 0898.46009 Fundam. Math. 153, No. 1, 81-98 (1997). Summary: It is shown that if \(\alpha\), \(\zeta\) are ordinals such that \(1\leq\zeta< \alpha<\zeta\omega\), 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 for any operator from \(C(\omega^{\omega^\alpha})\) onto itself there is a subspace of \(C(\omega^{\omega^\alpha})\) which is isomorphic to \(C(\omega^{\omega^\alpha})\) on which the operator is an isomorphism. Cited in 1 Document MSC: 46B03 Isomorphic theory (including renorming) of Banach spaces 06A07 Combinatorics of partially ordered sets 03E10 Ordinal and cardinal numbers Keywords:ordinal index; Szlenk index; Banach space of continuous functions PDFBibTeX XMLCite \textit{D. E. Alspach}, Fundam. Math. 153, No. 1, 81--98 (1997; Zbl 0898.46009) Full Text: arXiv EuDML