Into isomorphisms in tensor products of Banach spaces

Abstract

We establish quantitative extensions of two Grothendieck's results on into isomorphisms in projective tensor products. Among others, we prove the following. Let Y be a closed subspace of a Banach space Z and let j : Y → Z denote the identity embedding. If Y is complemented in its bidual Y**, then the injection modulus of the natural inclusion Id ⊗ j : Y*⊗Y → Y*⊗Z satisfies 1/λ _loc (Y,Z) ≤ i(Id ⊗ j) ≤ λ(Y,Y**)/λ(Y,Z), where λ(·,·) and λ_loc(·,·) are, respectively, the projection and the local projection constants.