Bidual Octahedral Renormings and Strong Regularity in Banach Spaces

Langemets, Johann; López-Pérez, Ginés

doi:10.1017/s1474748019000264

Cited by 8 publications

(2 citation statements)

References 13 publications

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“…It is also known that a Banach space E has an equivalent octahedral norm if and only if E contains an isomorphic copy of ℓ 1 (see [14]). Very recently, the previous result was improved in [22], where it was proved that if a Banach space E is separable and contains an isomorphic copy of ℓ 1 , then there exists an equivalent renorming of E such that the bidual norm is octahedral. Now, let us describe formally the objects we are working with in the paper.…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

Octahedral norms in free Banach lattices

Dantas¹,

Martínez-Cervantes²,

Abellán³

et al. 2020

Preprint

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In this paper, we study octahedral norms in free Banach lattices F BL [E] generated by a Banach space E. We prove that if E is an L 1 (µ)-space, a predual of von Neumann algebra, a predual of a JBW * -triple, the dual of an M -embedded Banach space, the disc algebra or the projective tensor product under some hypothesis, then the norm of F BL[E] is octahedral. We get the analogous result when the topological dual E * of E is almost square. We finish the paper by proving that the norm of the free Banach lattice generated by a Banach space of dimension ≥ 2 is nowhere Fréchet differentiable. Moreover, we discuss some open problems on this topic.

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Section: Notation and Preliminary Resultsmentioning

confidence: 99%

Octahedral norms in free Banach lattices

Dantas¹,

Martínez-Cervantes²,

Abellán³

et al. 2020

Preprint

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“…The natural norm of C([0, 1]) is octahedral, but its bidual norm is not. However, if X is a separable Banach space containing ℓ 1 , then there always exists an equivalent norm on X such that the bidual X * * is octahedral (see [21]). The non-separable case remains unknown as far as the authors know.…”

Section: Introductionmentioning

confidence: 99%

A characterization of Banach spaces containing $\ell_1(κ)$ via ball-covering properties

Ciaci,

Langemets,

Lissitsin

2021

Preprint

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In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of ℓ1 if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy's result to higher cardinalities. We prove that, for an infinite cardinal κ, a Banach space X contains an isomorphic copy of ℓ1(κ + ) if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by κ many open balls not containing αBX , where α ∈ (0, 1). We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.

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