Quotient maps with structure preserving inverses

It is proved that certain quotient maps q : (ℒ_n U_n)l₁ → Y, where U_n are finite dimensional spaces, have the following property: If E is a subspace of Y with a “good” structure of uniformly complemented finite dimensional subspaces, so is the subspace q⁻¹ (E) of (ℒ_n U_n)l₁. In particular, any quotient map q : l₁ → L₁ has this property.