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

Abstract

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.

Octahedral norms in tensor products of Banach spaces

“…Indeed, in [7, Theorem 2.5] it was proved that given two Banach spaces X and Y such that the norms of X * and Y are octahedral then the norm of every closed subspace H of L(X, Y ) which contains finite-rank operators is octahedral. As a corollary, the projective tensor product of two spaces having the SD2P enjoys the SD2P, a result which improved the main results of [4] and gave a partial answer to [3,Question (b)], where it was asked how diameter two properties are preserved by tensor product spaces. However, it remained an open problem whether the assumption of the SD2P on one of the factor can be eliminated [7, p. 177].…”

“…[7,Corollary 3.8] and Corollary 3.9 we see that, given two Banach spaces X and Y , it is not enough to assume that X has an infinite-dimensional centralizer to ensure that X ⊗ π Y has the SD2P. But both L ∞ and ℓ ∞ are isometric to C(K) spaces so L ∞ ⊗ π Y and ℓ ∞ ⊗ π Y do have the D2P for any Y by [4,Proposition 4.1].…”

Almost square and octahedral norms in tensor products of Banach spaces

“…From octahedrality we now turn to almost squareness and diameter two properties. Acosta, Becerra Guerrero and Rodríguez-Palacios have shown that if Y is a Banach space, then X ⊗ ε Y has the D2P whenever X is a Banach space such that the supremum of the dimension of the centralizer of all the even duals of X is unbounded [8,Theorem 5.3]. Now, we prove a stability result for ASQ, which will provide a wide class of injective tensor product spaces which have the SD2P.…”

“…for all x ∈ X (see the discussion following Theorem 3.6 in [8] for examples of spaces which have such unitaries). Note that n(X, u) = 1 if, and only if, D(X, u) is a norming subset for X.…”

Two remarks on diameter 2 properties

“…Nygaard and Werner [10] showed that in every infinite-dimensional uniform algebra, every nonempty relatively weakly open subset of its closed unit ball has diameter 2. If a Banach space satisfies this condition, then it is said to have the diameter 2 property (see, e.g., [1,3,5]). …”