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Operators on \(C(\omega^\alpha)\) which do not preserve \(C(\omega^\alpha)\). (English) Zbl 0898.46009

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.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
06A07 Combinatorics of partially ordered sets
03E10 Ordinal and cardinal numbers
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