Daniele Puglisi

Department of Mathematics and Computer Sciences
University of Catania
Viale Andrea Doria, 6
I-95125 Catania CT

Office: 346
Office phone: ++39 095 7383055
Fax: ++39 095 330094

E-mail: dpuglisiunict.it

Research

It was a long standing open question, raised by

• J. Diestel and J.J. Uhl, Vector Measures, In: Math. Surveys Vol. 15, Amer. Math. Soc., Providence, RI, 1977.

whether a tensor product (with a reasonable crossnorm) of two Banach spaces with the Radon-Nikodym property has the Radon-Nikodym property. In

• J. Bourgain and G. Pisier, A construction of L1-spaces and related Banach spaces, Bol. Soc. Brasil. Mat. 14(2) (1983), 109-123.

in a spectacular way, the authors constructed a Banach space X (still called the Bourgain-Pisier space) with the Radon-Nikodym property, such that, for every reasonable crossnorm, the tensor product of X with itself does not have the Radon-Nikodym property. In

• Daniele Puglisi; Fremlin tensor products of Banach lattices. Quaest. Math. 30 (2007), no. 1 45-56.

using descriptive set theory techniques, it is proved that for some particular reasonable crossnorm (the Fremlin tensor product), the above question of Diestel and Uhl holds true when we restrict in a Banach lattices context. Namely,

If U and X are two Banach lattices, one of them atomic, then the Fremlin tensor product of U and X, has the Radon-Nikodym property if both U and X possess this property.

Also, it is noted that the assumption of atomicity is sharp.

Questions

• If X is a Banach space with the l.u.st. property and Y is a Banach space with a basis, both with the Radon-Nikodym property, does the tensor product of X and Y has the Radon-Nikodym property relative to any reasonable crossnorm?
• What if X has the Gordon-Lewis property?

Comments

Keep in mind that, by

• T. Figiel, W.B. Johnson and L. Tzafriri; On Banach lattices and spaces having local unconditional structure with application to Lorentz function spaces, J. Approx. Theory 13 (1975), 297-312.

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