Banach Spaces of Continuous Functions as Dual Spaces

Dales, H. G.; Lau, Anthony To-Ming; Dashiell, F. K.; Strauss, Dona

doi:10.1007/978-3-319-32349-7

Cited by 51 publications

(56 citation statements)

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“…Then T is not β-to-β compact. By Corollary 4.5.9 of [12], the Banach space C(K) has the Grothendieck property, which means that the identity map C(K) ′ σ → (C(K) ′ β ) w is sequentially continuous. By Theorem 6.12, the Banach space C(K) does not have the eJNP.…”

Section: The Ejnp In Function Spacesmentioning

confidence: 98%

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Банах

Ka̧kol

Śliwa

2019

RACSAM

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We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E ′ , σ(E ′ , E)) → (E ′ , β * (E ′ , E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP.We show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak * null-sequence {µn}n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1(X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP. We also define two modifications of the JNP, called the universal JNP and the JNP everywhere (briefly, the uJNP and eJNP), and thoroughly study them in the classes of locally convex spaces, Banach spaces and function spaces. We provide a characterization of the JNP in terms of operators into locally convex spaces with the uJNP or eJNP and give numerous examples clarifying relationships between the considered notions.2010 Mathematics Subject Classification. Primary 46A03; Secondary 46E10, 46E15.

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Section: The Ejnp In Function Spacesmentioning

confidence: 98%

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Банах

Ka̧kol

Śliwa

2019

RACSAM

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“…Cole algebras. First recall that every C * -algebra A is Arens regular and that (A , ) is a C * -algebra [6, Theorem 3.2.36]; in particular, (C 0 (K) , ) is a commutative, unital C * -algebra, and so has the form C( K) for a compact space K, called the hyper-Stonean envelope of K. (See [7] for an extensive discussion and explicit constructions of the space K.) Now suppose that A is a uniform algebra on a non-empty, locally compact space K.…”

Section: Uniform Algebrasmentioning

confidence: 99%

Approximate identities in Banach function algebras

Dales¹,

Ülger²

2015

Studia Math.

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In this paper, we shall study contractive and pointwise contractive Banach function algebras, in which each maximal modular ideal has a contractive or pointwise contractive approximate identity, respectively, and we shall seek to characterize these algebras. We shall give many examples, including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras. We shall describe a contractive Banach function algebra which is not equivalent to a uniform algebra. We shall also obtain results about Banach sequence algebras and Banach function algebras that are ideals in their second duals.2010 Mathematics Subject Classification: Primary 46B15; Secondary 46B28, 46B42, 47L10.

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“…A regular Borel measure µ on X is called normal if ϕ µ (f i ) → 0 whenever f i ↓ 0 in C 0 (X); see [4,Definition 4.7.1]. This implies that ϕ µ is an order continuous linear functional on C 0 (X).…”

Section: Measure-theoretic Interpretationmentioning

confidence: 99%

On order continuous duals of vector lattices of continuous functions

Jeu

Walt

2019

Journal of Mathematical Analysis and Applications

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A topological space X is called resolvable if it contains a dense subset with dense complement. Using only basic principles, we show that the order continuous dual of a vector lattice of (not necessarily bounded) continuous functions on a resolvable space is trivial, whenever the space has a resolving subset that can be written as an at most countably infinite union of subsets, in such a way that the vector lattice separates every point outside the resolving subset from each of its constituents. We also show that several spaces have resolving subsets that can be written as at most countably infinite unions of closed nowhere dense subsets. Combination of the results yields, for example, that, under appropriate conditions, vector lattices of continuous functions on separable spaces, metric spaces, and topological vector spaces have trivial order continuous duals if they separate points and closed nowhere dense subsets. Our results in this direction extend known results in the literature. We also show that, under reasonably mild separation conditions, vector lattices of continuous functions on locally connected T 1 Baire spaces without isolated points have trivial order continuous duals. A discussion of the relation between our results and the non-existence of non-zero normal measures is included.

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Banach Spaces of Continuous Functions as Dual Spaces

Cited by 51 publications

References 0 publications

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Approximate identities in Banach function algebras

On order continuous duals of vector lattices of continuous functions

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