Some duality relations in the theory of tensor products

We study several classical duality results in the theory of tensor products, due mostly to Grothendieck, providing new proofs as well as new results. In particular, we show that the canonical mapping Y∗⊗πX→(L(X,Y),τ)∗Y∗⊗πX→(L(X,Y),τ)∗, where ττ is the topology of uniform convergence on compact subsets of X, is not always injective. This answers negatively a problem of Defant and Floret. We use the machinery of vector measures to give new proofs of the dualities (X⊗εY)∗=N(X,Y∗)(X⊗εY)∗=N(X,Y∗), whenever Y∗Y∗ has the Radon–Nikodým property, and (a slight improvement of) a result of Rosenthal: (X⊗εY)∗⊂F¯(X,Y∗) whenever ℓ1⁄↪Yℓ1⁄↪Y. Here, N(X,Y∗)N(X,Y∗) and F(X,Y∗)F(X,Y∗) denote the spaces of nuclear and finite rank operators from XX to Y∗Y∗, respectively.