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 λloc(·,·) are, respectively, the projection and the local projection constants.
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