A Lattice Renorming Theorem and Applications to Vector-Valued Processes

Davis, William J.; Ghoussoub, Nassif; Lindenstrauss, Joram

doi:10.2307/1998366

Cited by 14 publications

(20 citation statements)

References 4 publications

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“…The complex Banach space c 0 renormed by Day's norm is locally uniformly convex [19,20], but it doesn't have the Radon-Nikodým property [24]. In addition, it is a locally uniformly c-convex and order continuous Banach sequence space.…”

Section: We Next Choose Inductively Sequences {Fmentioning

confidence: 99%

Bishop’s theorem and differentiability of a subspace of C b (K)

2010

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Abstract. Let K be a Hausdorff space and C b (K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Fréchet differentiability of subspaces of C b (K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of C b (K) is a dense G δ subset of H, if K is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for H is the smallest closed norming subset of H. The classical Bishop's theorem was proved for a separating subalgebra H and a metrizable compact space K.In the case that X is a complex Banach space with the Radon-Nikodým property, we show that the set of all strong peak functions inX is holomorphic} is dense. As an application, we show that the smallest closed norming subset of A b (B X ) is the closure of the set of all strong peak points for A b (B X ). This implies that the norm of A b (B X ) is Gâteaux differentiable on a dense subset of A b (B X ), even though the norm is nowhere Fréchet differentiable when X is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary. IntroductionLet K be a Hausdorff topological space. A function algebra A on K will be understood to be a closed subalgebra of C b (K) which is the Banach algebra of all bounded complex-valued continuous functions on K. The norm f of a bounded continuous function f on K is defined to be sup x∈K |f (x)|. A function algebra A is called separating if for two distinct points s, t in K, there is f ∈ A such that f (s) = f (t).In this paper, a subspace means a closed linear subspace. For each t ∈ K, let δ t be an evaluation functional onis called separating if for distinct points t, s in K we have αδ t = βδ s for any complex numbers α, β of modulus 1 as a linear functional on A. This definition of a separating subspace is a natural extension of the definition of a separating function algebra. In fact, given a separating function algebra A 2000 Mathematics Subject Classification. 46B04, 46G20, 46G25, 46B22.

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Section: We Next Choose Inductively Sequences {Fmentioning

confidence: 99%

Bishop’s theorem and differentiability of a subspace of C b (K)

2010

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“…Therefore, by the Lebesgue dominated theorem, \\f* -f"*k II = f¿(f* -/*,)* w -0. Now, by a renorming result in [7], there exists a symmetric equivalent norm ||o in Aw that is locally uniformly rotund (for definition of local uniform rotundity, see e.g. [13]).…”

Section: /(O -G(t) < F(t)xa(t) + (/(F) -\F(t))xat) = /(F) -\F(t)xat)mentioning

confidence: 99%

Monotonicity properties of Lorentz spaces

Hudzik¹,

Kamińska²

1995

Proc. Amer. Math. Soc.

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Abstract. Criteria for uniform monotonicity, local uniform monotonicity and strict monotonicity of Lorentz spaces are given.Let L° denote the space of all (equivalence classes of ) Lebesgue measurable real-valued functions defined on the interval [0, y), y < oc. In what follows, if fyg £ L°, then f < g means f(t) < g(t) almost everywhere (a.e.) with respect to the Lebesgue measure m on the real line. If f £ L° we denote by df the distribution function of \f\, that is, df(t) = m{s:\f(s)\>t}, and we denote by /* the decreasing rearrangement of \f\, that is, f(t) = inf{5 > 0 : df(s) < t}.

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“…LEMMA (4). Let X be a separable subspace of the dual of a separable Banach space Y such that Bx is a w*-G¿ set which is w*-dense in By ■ Let L be a subset ofY* which is disjoint of X.…”

Section: The Asymptotic-norming and The Radon-nikodym Properties Are mentioning

confidence: 99%

“…We sketch an easier proof based on martingales and already used by Davis et al [4]. Let D be a countable dense set in the unit ball of Y.…”

Section: Corollary (5) (A)mentioning

confidence: 99%

The asymptotic-norming and the Radon-Nikodým properties are equivalent in separable Banach spaces

Ghoussoub¹,

Maurey²

1985

Proc. Amer. Math. Soc.

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ABSTRACT. We show that the asymptotic-norming and the Radon-Nikodym properties are equivalent, settling a problem of James and Ho [9]. In the process, we give a positive solution to two questions of Edgar and Wheeler [6] concerning Cech-complete Banach spaces. We also show that a separable Banach space with the Radon-Nikodym property semi-embeds in a separable dual whenever it has a norming space not containing an isomorphic copy of ii. This gives a partial answer to a problem of Bourgain and Rosenthal[3].Introduction. Let X be a Banach space. We recall that X is said to have the point of continuity property (resp. the Radon-Nikodym property) if every weakly closed bounded subset of X has a point of weak to norm continuity (resp. a denting point). A separable Banach space X is said to have the asymptotic norming property if there exists a separable Banach space Y such that X is (isomorphic to) a subspace of Y* which verifies the following property:

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A Lattice Renorming Theorem and Applications to Vector-Valued Processes

Cited by 14 publications

References 4 publications

Bishop’s theorem and differentiability of a subspace of C b (K)

Bishop’s theorem and differentiability of a subspace of C b (K)

Monotonicity properties of Lorentz spaces

The asymptotic-norming and the Radon-Nikodým properties are equivalent in separable Banach spaces

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