Almost square and octahedral norms in tensor products of Banach spaces

Abstract

Abstract. The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.

Daugavet Property in Tensor Product Spaces

“…In [1, Question b)] it is asked how are the diameter two properties, in general, preserved by tensor product spaces. Corollary 5.11 yields new examples of injetive tensor products with the slice-D2P different from those obtained in [2, Theorem 5.3] and of [22,Theorem 2.6].…”

Octahedral norms in tensor products of Banach spaces

“…In [21] it is shown that if X and Y are Banach spaces whose norms are octahedral then the norm of X ⊗ ε Y is also octahedral. The following proposition is similar to [ Proof.…”

“…The following theorem provides a partial positive answer to [21,Question 4.4], where it is asked whether octahedrality is preserved by taking projective tensor products from one of the factors. Theorem 4.3.…”

“…In Section 4 we study octahedrality of projective tensor products. In Theorem 4.3 we will prove that octahedrality is preserved from one of the factors by taking projective tensor products in presence of the metric approximation property whenever one of the factors in a non-reflexive L-embedded Banach space, which provides a partial positive answer to [21,Question 4.4].…”

On the structure of spaces of vector-valued Lipschitz functions

“…(2) Previous result has an immediate consequence in terms of octahedrality in Lipschitz-free Banach spaces. Recall that a Banach space X is said to have an octahedral norm if for every finite-dimensional subspace Y and for every ε > 0 there exists x ∈ S X verifying that y + λx > (1− ε)( y + |λ|) for every y ∈ Y and λ ∈ R. Notice that, given Banach spaces X and Y under the assumption of Proposition 2.11, it follows that F((B X * , ω • · ), Y * ) = F((B X * , ω • · )) ⊗ π Y * has an octahedral norm because of [19,Corollary 2.9]. Notice that this gives a partially positive answer to [3, Question 2], where it is wondered whether octahedrality in vector-valued Lipschitz-free Banach spaces actually relies on the scalar case.…”