Dugundji extension theorems for linearly ordered spaces

Heath, Robert W.; Lutzer, David

doi:10.2140/pjm.1974.55.419

Cited by 18 publications

(4 citation statements)

References 10 publications

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“…Let us remark that if K is a closed subset of a linearly ordered compact space L, then there is a regular extension operator E : C(K) → C(L), cf. [17]. Using this fact one can easily deduce Theorem 8.2 from Theorem 8.7, Lemma 8.6, Remark 2.5 and Proposition 2.6.…”

Section: Countable Discrete Extensions Of Dyadic Compactamentioning

confidence: 68%

Extension operators and twisted sums of c0 and C(K) spaces

Marciszewski

Plebanek

2018

Journal of Functional Analysis

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We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost:Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of c 0 and C(K), i.e., does there exist a Banach space X containing a non-complemented copy Z of c 0 such that the quotient space X/Z is isomorphic to C(K)?Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube 2 ω1 or K is a separable scattered compact space of height 3 and weight ω 1 , then every twisted sum of c 0 and C(K) is trivial.We also construct nontrivial twisted sums of c 0 and C(K) for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces K ⊆ L which do not admit an extension operator C(K) → C(L).

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Section: Countable Discrete Extensions Of Dyadic Compactamentioning

confidence: 68%

Extension operators and twisted sums of c0 and C(K) spaces

Marciszewski

Plebanek

2018

Journal of Functional Analysis

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“…Now it is known that Dugundji's extension theorem holds in some classes of generalized metric spaces X (C. J. R. Borges [3], I. S. Stares [11]), but does not hold for all GO-spaces X. Indeed, for the Michael line R Q , R. W. Heath and D. J. Lutzer [9] show that there exists no linear conv-extender u : C(Q) → C(R Q ); E. K. van Douwen [5] extends it by showing that there is no monotone extender u : C(Q) → C(R Q ) (see also I. S. Stares and J. E. Vaughan [12]). For related results on Dugundji extenders and retracts in GO-spaces, see G. Gruenhage, Y. Hattori and H. Ohta [8].…”

mentioning

confidence: 99%

“…On extenders for bounded functions, R. W. Heath and D. J. Lutzer [9] establish that for a closed subset A of a GO-space X, there exists a linear conv-extender u : C ∞ (A) → C ∞ (X); van Douwen's result [5] shows that "conv-extender" in the above cannot be strengthened to "conv-extender". For normed-space-valued functions, I. Banakh, T. Banakh and K. Yamazaki [1, Theorem 4.1] prove that a normed space Y is reflexive if and only if for every closed subset A of a GO-space X, there exists a linear convextender u :…”

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confidence: 99%

Monotone extenders for bounded c-valued functions

Yamazaki

2010

Studia Math.

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Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, C∞(A, c) the set of all bounded continuous functions f : A → c, and CA(X, c) the set of all functions f : X → c which are continuous at each point of A ⊂ X. We show that a Tikhonov subspace A of a topological space X is strong Choquet in X if there exists a monotone extender u : C∞(A, c) → CA(X, c). This shows that the monotone extension property for bounded c-valued functions can fail in GO-spaces, which provides a negative answer to a question posed by I. Banakh, T. Banakh and K. Yamazaki.

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“…An example due to Heath and Lutzer shows that this additional hypothesis is necessary: in [4] they show that if X is the Michael line [7] and A is the closed subspace of X consisting of all rational numbers, then no simultaneous extender from C(A*) to C(X) can be found.…”

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confidence: 99%

A Note on the Dugundji Extension Theorem

Lutzer¹,

Martin²

1974

Proceedings of the American Mathematical Society

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ABSTRACT.We prove that if A is a closed, metrizable, Gg-subspace of a collectionwise normal space X then there is a linear transformation e: C(A) -* C(X) such that for each g £ C(A), e(g) extends g and the range of e(g) is contained in the closed convex hull of the range of g. topology M on X having M C T.The second major ingredient in our proof is a slight generalization of the original Dugundji extension theorem wherein we consider pseudo-metriz-

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Dugundji extension theorems for linearly ordered spaces

Cited by 18 publications

References 10 publications

Extension operators and twisted sums of c0 and C(K) spaces

Extension operators and twisted sums of c0 and C(K) spaces

Monotone extenders for bounded c-valued functions

A Note on the Dugundji Extension Theorem

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