Copies of c 0 (τ ) spaces in projective tensor products

Vinícius Morelli Cortes (1) , Medina Galego (1) , Christian Samuel (2)

1 IME - Instituto de Matemática e Estatística
2 I2M - Institut de Mathématiques de Marseille

This paper is dedicated to the memory of our friend Eve Oja. Abstract. Let X and Y be Banach spaces and consider the projective tensor product X ⊗ π Y. Suppose that τ is an infinite cardinal, X has the bounded approximation property and the density character of X is strictly smaller than the cofinality of τ. We prove the following c 0 (τ) generalizations of classical c 0 results due to Oja (1991) and Kwapień (1974) respectively: (i) If c 0 (τ) is isomorphic to a complemented subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a complemented subspace of Y. (ii) If c 0 (τ) is isomorphic to a subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a subspace of Y. We also show that the result (i) is optimal for regular cardinals τ and Banach spaces X without copies of c 0 (τ). In order to do so, we provide a c 0 (τ) extension of a classical c 0 result due to Emmanuele (1988) concerning the c 0 (τ) complemented subspaces of Lp(D τ , Y) spaces, 1 ≤ p ≤ ∞, where D τ is the Cantor cube. Finally, as a consequence of (i) we conclude that under the continuum hypothesis, the space c 0 (ℵα), with α > 1, is not isomorphic to a complemented subspace of l∞ ⊗ π l∞(ℵα).

Domaines: Analyse fonctionnelle [math.FA]

Vinícius Morelli Cortes, Medina Galego, Christian Samuel. Copies of c 0 (τ ) spaces in projective tensor products. Proceedings of the American Mathematical Society, In press. ⟨hal-02886768⟩