In 1983 Bourgain and Pisier constructed, for every λ>1\lambda>1 and every separable Banach space EE, a separable L∞,λ\mathcal{L}_{\infty,\lambda}-space Lλ[E]\mathcal{L}_\lambda[E] containing EE isometrically, whose quotient Lλ[E]/E\mathcal{L}_\lambda[E]/E has the Radon–Nikodym property (RNP) and the Schur property; applying the construction to E=ℓ2E=\ell_2 produced a weakly sequentially complete space XX with the RNP whose completed projective tensor square X⊗^πXX\hat{\otimes}_\pi X contains an isomorphic copy of c0c_0 and therefore fails the RNP — a negative answer to a 1977 question of Diestel and Uhl. The one easy direction survives: the RNP of X⊗^πYX\hat{\otimes}_\pi Y forces the RNP of both factors (via the isometric embedding x↦x⊗y0x\mapsto x\otimes y_0 for a unit vector y0y_0), and the Bourgain–Pisier obstruction shows this necessary condition is not sufficient. Every later result is a partial sufficient condition: one factor with a boundedly complete 11-unconditional basis (Bu–Diestel–Dowling–Oja 2003), Köthe function-space factors (Bu–Lin 2004), the Fremlin projective tensor product of Banach lattices (Bu–Buskes 2006; Bu–Buskes–Lai 2008), and dual factors with the approximation property (Carando–Galicer 2011). No characterization of the pairs of RNP spaces whose projective tensor product retains the RNP — nor of the tensor norms between the injective and projective norms that preserve it — is known in the literature as of 2026. The objective below is that open core.

Background

The Radon–Nikodym property (RNP) is a central regularity property of Banach spaces. A Banach space XX has the RNP if every XX-valued countably additive measure of bounded variation that is absolutely continuous with respect to a positive measure admits a Bochner-integrable density (Bochner integrability being the standard generalization of Lebesgue integration to vector-valued functions); equivalently, if every XX-valued martingale (a sequence of random variables adapted to an increasing filtration) that is bounded in L1L^1 converges in norm. Reflexive spaces and separable dual spaces have the RNP, while c0c_0 — the scalar sequences converging to 00 under the supremum norm — fails it; the RNP passes to closed subspaces. The Schur property is the sequential condition that every weakly convergent sequence in the space is norm convergent; a space with the Schur property is weakly sequentially complete and contains no copy of c0c_0, since there every weakly Cauchy sequence is norm Cauchy and hence norm convergent. A space is weakly sequentially complete if every weakly Cauchy sequence has a weak limit; this excludes c0c_0.

An L∞\mathcal{L}_\infty-space is a Banach space whose finite-dimensional subspaces are uniformly close to finite-dimensional ℓ∞n\ell_\infty^n spaces (ℓ∞n\ell_\infty^n being Rn\mathbb{R}^n or Cn\mathbb{C}^n with the supremum norm): XX is an L∞,λ\mathcal{L}_{\infty,\lambda}-space, λ≥1\lambda\ge 1, if every finite-dimensional subspace of XX is contained in a finite-dimensional subspace whose Banach–Mazur distance to some ℓ∞n\ell_\infty^n is at most λ\lambda — the Banach–Mazur distance between two isomorphic spaces being inf⁡{∥T∥ ∥T−1∥}\inf\{\|T\|\,\|T^{-1}\|\} over all linear isomorphisms TT — and XX is an L∞\mathcal{L}_\infty-space if it is an L∞,λ\mathcal{L}_{\infty,\lambda}-space for some finite λ\lambda. These are the linear relatives of C(K)C(K)-spaces (continuous functions on compact Hausdorff spaces). Bourgain and Delbaen (1980) produced the first separable L∞\mathcal{L}_\infty-space with the Schur property — in particular one without any copy of c0c_0.

Bourgain and Pisier (1983) then proved the embedding theorem: for λ>1\lambda>1 and any separable Banach space EE there is a separable L∞,λ\mathcal{L}_{\infty,\lambda}-space Lλ[E]\mathcal{L}_\lambda[E] containing EE isometrically, such that the quotient Lλ[E]/E\mathcal{L}_\lambda[E]/E has both the RNP and the Schur property; the construction generalizes that of Bourgain and Delbaen. The engine is an inductive iteration of finite-dimensional extensions, governed by two notions. An isometric embedding j:F→Gj:F\to G of finite-dimensional spaces is η\eta-admissible, for 1/λ<η<11/\lambda<\eta<1, when there exist a Banach space BB and a metric surjection (a surjective linear map realizing the quotient norm) π:B⊕1F→G\pi:B\oplus_1 F\to G with π(0,f)=j(f)\pi(0,f)=j(f) for all f∈Ff\in F and ∥π(b,f)∥≥∥f∥−η∥b∥\|\pi(b,f)\|\ge\|f\|-\eta\|b\| for all b∈Bb\in B and f∈Ff\in F; informally, GG is presented as a quotient of an ℓ1\ell_1-sum extending jj in which the added summand can erode the norm of the FF-component only at the rate η<1\eta<1. A closed subspace HH of a Banach space ZZ is δ\delta-well-placed, for 0<δ<10<\delta<1, when the quotient map q:Z→Z/Hq:Z\to Z/H satisfies the averaged inequality E∥z∥≥∥Ez∥+δ E∥q(z)∥\mathbb{E}\|z\|\ge\|\mathbb{E}z\|+\delta\,\mathbb{E}\|q(z)\| for every ZZ-valued random variable zz of finite first moment with Ez∈H\mathbb{E}z\in H. The two notions are linked by δ=(1−η)/(1+η)\delta=(1-\eta)/(1+\eta): an η\eta-admissible embedding makes the copy j(F)j(F) δ\delta-well-placed in GG, and the placement propagates through inductive limits. Iterating η\eta-admissible extensions along an increasing sequence of finite-dimensional subspaces with dense union in EE and passing to the inductive limit yields Lλ[E]\mathcal{L}_\lambda[E]; the propagated δ\delta-well-placedness of EE inside the limit is the mechanism by which the quotient Lλ[E]/E\mathcal{L}_\lambda[E]/E acquires the Schur property and the RNP.

Two corollaries of the construction are the historical anchors here. First, for each λ>1\lambda>1 there is an L∞,λ\mathcal{L}_{\infty,\lambda}-space failing the RNP yet containing no copy of c0c_0, answering a question raised on p. 46 of Bourgain's 1981 Lecture Notes (New Classes of Lp\mathcal{L}_p-spaces). Second — the anchor of the present problem — for each λ>1\lambda>1 there is an L∞,λ\mathcal{L}_{\infty,\lambda}-space XX, obtained by applying the construction to E=ℓ2E=\ell_2, which is weakly sequentially complete and has the RNP, while its completed projective tensor square X⊗^πXX\hat{\otimes}_\pi X contains an isomorphic copy of c0c_0 and hence fails the RNP. Here ⊗^π\hat{\otimes}_\pi denotes completion under the projective tensor norm ∥u∥∧=inf⁡{∑i∥xi∥ ∥yi∥:u=∑ixi⊗yi}\|u\|_\wedge=\inf\{\sum_i\|x_i\|\,\|y_i\|: u=\sum_i x_i\otimes y_i\}, the largest reasonable crossnorm on X⊗YX\otimes Y; the injective tensor norm is ∥u∥∨=sup⁡{∣(x∗⊗y∗)(u)∣:∥x∗∥≤1,∥y∗∥≤1}\|u\|_\vee=\sup\{|(x^*\otimes y^*)(u)|: \|x^*\|\le 1,\|y^*\|\le 1\}, the smallest one, and a reasonable crossnorm is any norm α\alpha on X⊗YX\otimes Y, compatible with the dual pairing, with ∥⋅∥∨≤∥⋅∥α≤∥⋅∥∧\|\cdot\|_\vee\le\|\cdot\|_\alpha\le\|\cdot\|_\wedge. Writing (ei)(e_i) for the canonical basis of ℓ2⊂X\ell_2\subset X and KGK_G for the constant in Grothendieck's inequality (which bounds bilinear forms on products of ℓ∞n\ell_\infty^n-spaces, and up to the factor λ\lambda on products of L∞,λ\mathcal{L}_{\infty,\lambda}-spaces, by an ℓ2\ell_2-type estimate), the two-sided estimate

sup⁡1≤i≤N∣αi∣ ≤ ∥∑i=1Nαi ei⊗ei∥∧ ≤ λ KG sup⁡1≤i≤N∣αi∣\sup_{1\le i\le N}|\alpha_i|\ \le\ \Big\|\sum_{i=1}^N \alpha_i\, e_i\otimes e_i\Big\|_\wedge\ \le\ \lambda\,K_G\,\sup_{1\le i\le N}|\alpha_i|

shows that the closed span of the diagonal tensors (ei⊗ei)(e_i\otimes e_i) in X⊗^πXX\hat{\otimes}_\pi X is isomorphic to c0c_0: the lower bound comes from the injective norm (the coordinate functionals of ℓ2\ell_2, extended by Hahn–Banach, evaluate the diagonal), the upper bound from Grothendieck's inequality. Since the RNP passes to subspaces and c0c_0 fails the RNP, X⊗^πXX\hat{\otimes}_\pi X fails the RNP. The same two-sided estimate persists under any reasonable crossnorm α\alpha between the injective and projective norms, so X⊗^αXX\hat{\otimes}_\alpha X contains c0c_0 as well. This answered negatively the question, posed on p. 258 of the 1977 Diestel–Uhl monograph, whether the RNP of two factors implies the RNP of their projective tensor product.

The one easy direction survives: for any unit vector y0∈Yy_0\in Y the map x↦x⊗y0x\mapsto x\otimes y_0 embeds XX isometrically into X⊗^πYX\hat{\otimes}_\pi Y, so the RNP of X⊗^πYX\hat{\otimes}_\pi Y forces the RNP of both factors. The Bourgain–Pisier counterexample shows this necessary condition is not sufficient. What is missing — and what the later literature has only chipped away at — is the boundary in between: an identifiable structural feature of a pair of RNP spaces that is equivalent to the RNP of their completed projective tensor product, and the corresponding division of reasonable crossnorms into those that destroy and those that preserve the property.

Problem Statement

Determine the tensor-stability boundary of the Radon–Nikodym property (RNP) for Banach spaces, as left open by the 1983 Bourgain–Pisier counterexample. Precisely:

(1) Characterize the pairs (X,Y)(X,Y) of Banach spaces with the RNP whose completed projective tensor product X⊗^πYX\hat{\otimes}_\pi Y also has the RNP — that is, identify a structural feature of the pair that is necessary and sufficient, given that the RNP of both factors is necessary (the isometric embedding x↦x⊗y0x\mapsto x\otimes y_0 for any unit vector y0y_0) yet not sufficient (the space X=Lλ[ℓ2]X=\mathcal{L}_\lambda[\ell_2] of Bourgain and Pisier has the RNP and is weakly sequentially complete, while X⊗^πXX\hat{\otimes}_\pi X contains an isomorphic copy of c0c_0).

(2) Determine, at the level of tensor norms, for which reasonable crossnorms α\alpha with ∥⋅∥∨≤∥⋅∥α≤∥⋅∥∧\|\cdot\|_\vee\le\|\cdot\|_\alpha\le\|\cdot\|_\wedge on X⊗YX\otimes Y the completed product X⊗^αYX\hat{\otimes}_\alpha Y retains the RNP. The universal reading — a single such α\alpha for which the RNP of both factors would always suffice — is settled negatively: the Bourgain–Pisier diagonal argument applies to every such α\alpha in the square case X=Y=Lλ[ℓ2]X=Y=\mathcal{L}_\lambda[\ell_2]. The open content is the pair-relative boundary: for which pairs (X,Y)(X,Y) of RNP spaces do some or all intermediate crossnorms preserve the RNP, and how does the set of preserving norms depend on the pair.

Both parts (1) and (2) are posed jointly.

The objective stated as (1) and (2) is the core of the research question attached to the 1983 paper, which also raised the limiting behavior of the construction's parameter η↑1\eta\uparrow 1 toward the Gurarij space and a classification of the spaces Lλ[E]\mathcal{L}_\lambda[E]. Those two lines are outside the scope of this task.

References

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