A complementary universal conjugate Banach space and its relation to the approximation problem

LetC ₁=(ΣG _n ) _{l 1}, where (G _n ) is a sequence which is dense (in the Banach-Mazur sense) in the class of all finite dimensional Banach spaces. IfX is a separable Banach space, thenX ^* is isometric to a subspace ofC ₁ ^* =(ΣG _n ^* ) _m which is the range of a contractive projection onC ₁ ^* . Separable Banach spaces whose conjugates are isomorphic toC ₁ ^* are classified as those spaces which contain complemented copies of C₁. Applications are that every Banach space has the [metric] approximation property ([m.] a.p., in short) iff (ΣG _n ^* ) _m does, and if there is a space failing the m.a.p., thenC ₁ can be equivalently normed to fail the m.a.p. The author was supported in part by NSF GP 28719.