Operators on $C(ω^α)$ which do not preserve $C(ω^α)$
Fundamenta Mathematicae, Tome 153 (1997) no. 1, pp. 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.
Keywords:
ordinal index, Szlenk index, Banach space of continuous functions
@article{10_4064_fm_153_1_81_98,
author = {Dale E. Alspach},
title = {Operators on $C(\ensuremath{\omega}^\ensuremath{\alpha})$ which do not preserve $C(\ensuremath{\omega}^\ensuremath{\alpha})$},
journal = {Fundamenta Mathematicae},
pages = {81--98},
year = {1997},
volume = {153},
number = {1},
doi = {10.4064/fm-153-1-81-98},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm-153-1-81-98/}
}
Dale E. Alspach. Operators on $C(ω^α)$ which do not preserve $C(ω^α)$. Fundamenta Mathematicae, Tome 153 (1997) no. 1, pp. 81-98. doi: 10.4064/fm-153-1-81-98
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