On dominated extensions in linear subspaces of 𝒞C(X)

Alfsen, Erik M.; Hirsberg, Bent

doi:10.2140/pjm.1971.36.567

Cited by 25 publications

(23 citation statements)

References 11 publications

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“…1 , but then -Ts^Γα*)^ e JB- 1 . We can then using Theorems 1.7 and 1.9 deduce the same interpolation theorems as in [2] and [3]. In particular we get from Theorem 1.…”

Section: B\ F Is Closed In C(f)mentioning

confidence: 59%

Normpreserving extensions in subspaces ofC(X)

Briem¹,

Rao²

1972

Pacific J. Math.

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If B is a subspace of C(X) and F is a closed subset of X, this note gives sufficient conditions in order that every function in the restriction subspace B\ F has an extension in B with no increase in norm.Introduction* Let X be a compact Hausdorff space, C{X) the Banach algebra of all continuous complex-valued functions on X and let B be a closed linear subspace of C(X) separating the points of X and containing the constants. A closed subset F of X is said to have the normpreserving extension property w r.t. B if any function b Q in the restriction subspace B\ F has an extension beB (i.e. b\ F = b Q ) such that II&H = ||6 O |U(II 11(resp. || \\ F ) denotes the supremum norm on X (resp. F)).The main result is the following:

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“…1 , but then -Ts^Γα*)^ e JB- 1 . We can then using Theorems 1.7 and 1.9 deduce the same interpolation theorems as in [2] and [3]. In particular we get from Theorem 1.…”

Section: B\ F Is Closed In C(f)mentioning

confidence: 59%

Normpreserving extensions in subspaces ofC(X)

Briem¹,

Rao²

1972

Pacific J. Math.

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“…In known generalizations and applications of Bishop's theorem [1], [3], [4], [5], [6], [7], [9], [10], [11] the dominating functions q are always assumed to be positive on K. Our purpose here is to present a special case where the conclusion of Bishop's theorem can be strengthened to allow domination by nonnegative functions.…”

mentioning

confidence: 99%

On dominated extensions in function algebras

Globevnik¹

1980

Proc. Amer. Math. Soc.

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The Bishop-Gamelin interpolation theorem asserts that given a compact Hausdorff space K, a closed subspace A of C ( K ) C(K) , a positive continuous function p on K and a closed set F ⊂ K F \subset K such that every measure in the annihilator of A vanishes on F, every function f ∈ C ( F ) f \in C(F) satisfying | f ( s ) | ⩽ p ( s ) ( s ∈ F ) |f(s)| \leqslant p(s)(s \in F) extends to a function f ~ ∈ A \tilde f \in A satisfying | f ~ ( z ) | ⩽ p ( z ) ( z ∈ K ) |\tilde f(z)| \leqslant p(z)(z \in K) . In the paper we consider a special case where the theorem is extended to the situation when the dominating function is nonnegative.

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“…The construction of the finite dimensional extensions involves among others ideas from a paper of Rao [11] and the construction is actually reduced to showing the existence of an extension of a single function on a new M-set in a space larger than X, the proof of the existence of this extension is then a slight modification of the proofs in the papers of Alfsen and Hirsberg [3] and the author and Rao [7].…”

mentioning

confidence: 99%

“…In [9] it is proved that a closed subspace J of F is an M-ideal if and only if there exists an M-set F for F such that J = {b G B: b\F = 0). A slight modification of the proofs in [3] or [7] shows that if F is an M-set then each function in B\F has a norm preserving extension to a function in B. Also, the F-sets are generalized peak sets (i.e.…”

mentioning

confidence: 99%

“…Also, the F-sets are generalized peak sets (i.e. intersections of peak sets [3,Theorem 5.4 and remark]) so that the M-sets might alternatively be characterized as those generalized peak sets F for F for which ju|f E Fx for all H G F-1 n M(dBX). We note that conditions (i) and (iii) in the definition of M-sets may be replaced by the condition that co (F), the closed convex hull of F in SB, is a face of^Sg with F = X n co (F).…”

mentioning

confidence: 99%

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