  There exist injective Tauberian operators on $L_1(0,1)$ that have dense, non
closed range. This gives injective, non surjective operators on $\ell_\infty$
that have dense range. Consequently, there are two quasi-complementary, non
complementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$.
