On weak${}^{\ast }$-extensible subspaces of Banach spaces

Abstract

Let $X$ be a Banach space and $Y\subseteq X$ be a closed subspace. We prove that if the quotient $X/Y$ is weakly Lindelöf determined or weak Asplund, then for every $w^{\ast }$-convergent sequence $(y_n^{\ast }{)}_{n\in \mathbb{N}}$ in $Y^{\ast }$ there exist a subsequence $(y_{n_k}^{\ast }{)}_{k\in \mathbb{N}}$ and a $w^{\ast }$-convergent sequence $(x_k^{\ast }{)}_{k\in \mathbb{N}}$ in $X^{\ast }$ such that $x_k^{\ast }{|}_Y=y_{n_k}^{\ast }$ for all $k\in \mathbb{N}$. 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.