A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces

AbstractLet λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ∞λ-spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in [2]. Various choices ofE allow us to answer several questions raised in the literature. In particular, takingE = ℓ2, we obtain a ℒ∞λ-spaceX with the RNP such that the projective tensor product $$X\hat \otimes X$$ containsc0 and hence fails the RNP. TakingE=L1, we obtain a ℒ∞λ-space failing the RNP but nevertheless not containingc0.