# 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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topAlspach, 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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- [B] J. Bourgain, The Szlenk index and operators on C(K)-spaces, Bull. Soc. Math. Belg. Sér. B 31 (1979), 87-117. Zbl0438.46014
- [G] I. Gasparis, Quotients of C(K) spaces, dissertation, The University of Texas, 1995.
- [G1] I. Gasparis, Operators that do not preserve C(α)-spaces, preprint.
- [MS] S. Mazurkiewicz et W. Sierpiński, Contributions à la topologie des ensembles dénombrables, Fund. Math. 1 (1920), 17-27. Zbl47.0176.01
- [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
- [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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