$\Sigma$-completeness and closed graph theorems

A certain completeness concept, $\Sigma$-completeness, allows us to give some closed graph theorems between spaces whose dual space is weak* $\Sigma$-complete and some classes of Banach spaces in the range space, namely: reflexive Banach spaces, Banach spaces without copy of $\ell_\infty$ and dual spaces of Banach spaces with unconditional basis and without copy of $\ell_1$