Banach spaces whose duals contain complemented subspaces isomorphic to C[0, 1]

We give several characterizations of those Banach spaces X such that the dual $\text{X}{}^1$ contains a complemented subspace isomorphic to $\text{C0, 1]}{}^1$. We investigate operators on separable $\text{L}$_∞ spaces whose adjoints have nonseparable ranges and apply our results to obtain a structure theorem for $\text{L}$_∞ spaces whose duals are not isomorphic to l₁(Γ).