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

Volume 153 / 1997

Dale E. Alspach Fundamenta Mathematicae 153 (1997), 81-98 DOI: 10.4064/fm-153-1-81-98


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.


  • Dale E. Alspach

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