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

Fundamenta Mathematicae (1997)

• Volume: 153, Issue: 1, page 81-98
• ISSN: 0016-2736

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Abstract

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It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself such that if Y is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$, 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 $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself there is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$ on which the operator is an isomorphism.

How to cite

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Alspach, Dale. "Operators on C(ω^α) which do not preserve C(ω^α)." Fundamenta Mathematicae 153.1 (1997): 81-98. <http://eudml.org/doc/212216>.

@article{Alspach1997,
abstract = {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.},
author = {Alspach, Dale},
journal = {Fundamenta Mathematicae},
keywords = {ordinal index; Szlenk index; Banach space of continuous functions},
language = {eng},
number = {1},
pages = {81-98},
title = {Operators on C(ω^α) which do not preserve C(ω^α)},
url = {http://eudml.org/doc/212216},
volume = {153},
year = {1997},
}

TY - JOUR
AU - Alspach, Dale
TI - Operators on C(ω^α) which do not preserve C(ω^α)
JO - Fundamenta Mathematicae
PY - 1997
VL - 153
IS - 1
SP - 81
EP - 98
AB - 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.
LA - eng
KW - ordinal index; Szlenk index; Banach space of continuous functions
UR - http://eudml.org/doc/212216
ER -

References

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1. [A1] D. E. Alspach, Quotients of C[0,1] with separable dual, Israel J. Math. 29 (1978), 361-384. Zbl0375.46027
2. [A2] D. E. Alspach, C(K) norming subsets of C[0,1]*, Studia Math. 70 (1981), 27-61. Zbl0387.46026
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5. [B] J. Bourgain, The Szlenk index and operators on C(K)-spaces, Bull. Soc. Math. Belg. Sér. B 31 (1979), 87-117. Zbl0438.46014
6. [G] I. Gasparis, Quotients of C(K) spaces, dissertation, The University of Texas, 1995.
7. [G1] I. Gasparis, Operators that do not preserve C(α)-spaces, preprint.
8. [MS] S. Mazurkiewicz et W. Sierpiński, Contributions à la topologie des ensembles dénombrables, Fund. Math. 1 (1920), 17-27. Zbl47.0176.01
9. [P] A. Pełczyński, On strictly singular and cosingular operators I. Strictly singular and strictly cosingular operators on C(S) spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1965), 31-36. Zbl0138.38604
10. [W] J. Wolfe, C(α) preserving operators on C(K) spaces, Trans. Amer. Math. Soc. 273 (1982), 705-719. Zbl0526.46031

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