On Weak$$^*$$-Extensible Subspaces of Banach Spaces

Abstract

AbstractLet X be a Banach space and $$Y \subseteq X$$ Y ⊆ X be a closed subspace. We prove that if the quotient X/Y is weakly Lindelöf determined or weak Asplund, then for every $$w^*$$ w ∗ -convergent sequence $$(y_n^*)_{n\in \mathbb N}$$ ( y n ∗ ) n ∈ N in $$Y^*$$ Y ∗ there exist a subsequence $$(y_{n_k}^*)_{k\in \mathbb N}$$ ( y n k ∗ ) k ∈ N and a $$w^*$$ w ∗ -convergent sequence $$(x_k^*)_{k\in \mathbb N}$$ ( x k ∗ ) k ∈ N in $$X^*$$ X ∗ such that $$x_k^*|_Y=y_{n_k}^*$$ x k ∗ | Y = y n k ∗ for all $$k\in \mathbb N$$ k ∈ N . As an application, we obtain that Y is Grothendieck whenever X is Grothendieck and X/Y is reflexive, which answers a question raised by González and Kania.