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