On ${\ell }_{\infty }$-Grothendieck subspaces

Abstract

A closed subspace $S$ of ${\ell }_{\infty }$ is said to be a \emph{${\ell }_{\infty }$-Grothendieck subspace} if $c_0\subset S$ (hence ${\ell }_{\infty }\subset S^{\ast \ast }$) and every $\sigma (S^{\ast },S)$-convergent sequence in $S^{\ast }$ is $\sigma (S^{\ast },{\ell }_{\infty })$-convergent. Here we give examples of closed subspaces of ${\ell }_{\infty }$ containing $c_0$ which are or fail to be ${\ell }_{\infty }$-Grothendieck.