Examples of separable spaces which do not contain $l_{1}$ and whose duals are non-separable

Lindenstrauss, Joram; Stegall, Charles

doi:10.4064/sm-54-1-81-105

Cited by 106 publications

(77 citation statements)

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“…For example, it is not true that every nonseparable subspace of L°°{0, 1}N° contains a subspace isomorphic to I1. For by a result of R. C. James [7], [9], there exists a separable Banach space X with X* nonseparable but such that neither X nor any successive dual of X contains a subspace isomorphic to I1. But since the dual of every separable space is isomorphic to a subspace of L°°{0, 1}H°, there is an isomorph of X* in L°°{0, 1}X° which does not contain ll.…”

Section: Corollarymentioning

confidence: 99%

On the structure of 𝑆 and 𝐶(𝑆) for 𝑆 dyadic

Hagler¹

1975

Trans. Amer. Math. Soc.

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ABSTRACT. A dyadic space S is defined to be a continuous image of {o, l}m for some infinite cardinal number m. We deduce Banach space properties of C(S) and topological properties of S. For example, under certain cardinality restrictions on m, we show: Every dyadic space of topological weight m contains a closed subset homeomorphic to {o, l}m. Every Banach space X isomorphic to an m dimensional subspace of C(S) (for S dyadic) contains a subspace isomorphic to /'(r) where r has cardinality m.Introduction. In this paper we examine the relationships between dyadic spaces S and the Banach spaces C(S) of continuous real valued functions on S. The main result is

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Section: Corollarymentioning

confidence: 99%

On the structure of 𝑆 and 𝐶(𝑆) for 𝑆 dyadic

Hagler¹

1975

Trans. Amer. Math. Soc.

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“…The first examples were constructed independently by James [21] and Lindenstrauss and Stegall [27]. In view of Theorem 1.3 we now see that a fruitful way of producing such distinguishing examples comes from dynamical systems.…”

Section: ) (G X) Is a Tame G-system (2) (G X) Is Representable Onmentioning

confidence: 99%

Representations of dynamical systems on Banach spaces not containing $l_{1}$

Glasner

Megrelishvili

2012

Trans. Amer. Math. Soc.

107

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“…The James tree space JT (see [14]) is simple. Let us sketch a proof of this: JT * * = JT ⊕ 2 (Γ), where Γ is the set of all branches of a dyadic tree.…”

Section: Spaces Not Containing Cmentioning

confidence: 99%