NONSEPARABLE COMPLEMENTED SUBLATTICES IN THE BANACH ENVELOPE OF $WeakL_l$

We investigate complemented Banach sublattices of the Banach envelope of $Weak_L1$. In particular, the Banach envelope of $Weak_L1$ contains a complemented Banach sublattice that is isometrically isomorphic to a nonseparable Banach lattice $l_p(S),\;1{\leq}p<{\infty}\;and\;|S|{\leq}2^{{\aleph}0}$.