Relatively weakly open sets in closed balls of Banach spaces, and the centralizer

A Banach space is said to have the diameter two property if every non-empty relatively weakly open subset of its unit ball has diameter two. We prove that the projective tensor product of two Banach spaces whose centralizer is infinite-dimensional has the diameter two property. The same statement also holds for X ⊗ π Y if the centralizer of X is infinitedimensional and the unit sphere of Y * contains an element of numerical index one.We provide examples of classical Banach spaces satisfying the assumptions of the results.If K is any infinite compact Hausdorff topological space, then C(K ) ⊗ π Y has the diameter two property for any nonzero Banach space Y . We also provide a result on the diameter two property for the injective tensor product.

show abstract

“…Indeed, it is known that the image of Z (X) under this embedding is contained in Z (X (∞ ) (see [10,Proposition 4.3]). …”

Section: Lemma 25 Let X and Y Be Banach Spaces And Assume That Z (Xmentioning

confidence: 97%

“…In the case of the injective tensor product, it will be enough to assume one restriction only to one of the spaces in order to obtain a positive result, as we will see later. [10,Theorem 4.4].) Let X be a Banach space failing the diameter two property.…”

Section: Injective Tensor Productmentioning

confidence: 99%

Weakly open sets in the unit ball of the projective tensor product of Banach spaces

Acosta

Guerrero

Rodríguez-Palacios

2011

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…Afterwards, we give the necessary definitions and background in order to prove Theorem 3.1. We follow the notation from [9]; given a Banach space E, we denote by C(E) the Cunningham algebra of E (see the paragraph preceding Theorem 3.3 for a formal definition) and by E (∞ the completion of the normed space ∞ n=0 E (2n , where (E (2n ) ∞ n=0 is the sequence of even duals such that E ⊆ E * * ⊆ E (4 ⊂ . .…”

Section: Octahedral Norms In Free Banach Lattices In Terms Of the Cun...mentioning

confidence: 99%

Octahedral norms in free Banach lattices

Dantas¹,

Martínez-Cervantes²,

Abellán³

et al. 2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…Let p be an extreme point of

B_{X}

. Then

∥ p (t) ∥ = 1, \forall t \in K,

by [, Lemma 2.1]. Consider the analytic mapping given by

{(λ_{i})}_{1}^{n} \in C^{n} \overset{L_{p}}{\mapsto} x + \frac{1}{2} T ({(λ_{i})}_{1}^{n}) p \in X .

We claim that

∥ L_{p} ({(λ_{i})}_{1}^{n} ∥ < 1

∥ {(λ_{i})}_{1}^{n} ∥_{\infty} \leq 1 .

Indeed, if

t \in K,

then either there exists a unique

k \in S

such that

t \in O_{k}

t \in K ∖ ⋃_{i \in S} O_{i} .

t \in O_{k},

we have that

∥ L_{p} ({(λ_{i})}_{1}^{n}) (t) ∥ \leq ∥ x (t) ∥ + \frac{1}{2} | λ_{k} | m_{k} (t) ∥ p (t) ∥ \leq \frac{1}{3} + \frac{1}{2} < 1 .

While if

t \in K ∖ ⋃_{i \in S} O_{i},

then

∥ L_{p} ((λ_{i})...

…”

Section: Banach Function Modulesmentioning

confidence: 99%

Boundaries for algebras of analytic functions on function module Banach spaces

Acosta

Galindo

Lourenço

2013

Mathematische Nachrichten

View full text Add to dashboard Cite

We consider the uniform algebra A ∞ (B X ) of continuous and bounded functions that are analytic on the interior of the closed unit ball B X of a complex Banach function module X. We focus on norming subsets of B X , i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinitedimensional, then the intersection of all closed boundaries is empty. This also holds in case that X is an ∞ -sum of infinitely many Banach spaces and further, the torus is a boundary.

show abstract

Relatively weakly open sets in closed balls of Banach spaces, and the centralizer

Cited by 7 publications

References 21 publications

Weakly open sets in the unit ball of the projective tensor product of Banach spaces

Weakly open sets in the unit ball of the projective tensor product of Banach spaces

Octahedral norms in free Banach lattices

Boundaries for algebras of analytic functions on function module Banach spaces

Contact Info

Product

Resources

About