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

• Autorzy: Dale E. Alspach
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

ordinal index; Szlenk index; Banach space of continuous functions

Kategorie tematyczne

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

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1997
• Tom: 153
• Numer: 1
• Strony: 81-98

Daty

wydano

1997

otrzymano

1996-10-22

Twórcy

autor

Dale E. Alspach

• Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078, U.S.A.

Bibliografia

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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.
• [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.
• [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.
• [W] J. Wolfe, C(α) preserving operators on C(K) spaces, Trans. Amer. Math. Soc. 273 (1982), 705-719.