# The projective tensor product (II): the Radon-Nikodym property.

Joe Diestel; Jan Fourie; Johan Swart

RACSAM (2006)

- Volume: 100, Issue: 1-2, page 75-100
- ISSN: 1578-7303

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topDiestel, Joe, Fourie, Jan, and Swart, Johan. "The projective tensor product (II): the Radon-Nikodym property.." RACSAM 100.1-2 (2006): 75-100. <http://eudml.org/doc/41644>.

@article{Diestel2006,

abstract = {In this paper we discuss the problem of when the projective tensor product of two Banach spaces has the Radon-Nikodym property. We give a detailed exposition of the famous examples of Jean Bourgain and Gilles Pisier showing that there are Banach spaces X and Y such that each has the Radon-Nikodym property but for which their projective tensor product does not; this result depends on the classical theory of absolutely summing, integral and nuclear operators, as well as the famous Grothendieck inequality for its punch-line. In the last section of this paper we discuss many results of a positive character, due to Qingying Bu and various of his coauthors; in particular, we mention results of Bu, Diestel, Dowling and Oja to the effect that if one of the spaces has a boundedly complete FDD then the projective tensor product of two spaces with the RNP has it and a modification of a result of Bu and Pei-Kee Lin to the effect that if X is a Banach lattice with RNP and Y is any Banach space with RNP then their projective product has RNP.},

author = {Diestel, Joe, Fourie, Jan, Swart, Johan},

journal = {RACSAM},

keywords = {projective tensor product; Radon-Nikodym property; Bourgain-Pisier space; Banach lattice},

language = {eng},

number = {1-2},

pages = {75-100},

title = {The projective tensor product (II): the Radon-Nikodym property.},

url = {http://eudml.org/doc/41644},

volume = {100},

year = {2006},

}

TY - JOUR

AU - Diestel, Joe

AU - Fourie, Jan

AU - Swart, Johan

TI - The projective tensor product (II): the Radon-Nikodym property.

JO - RACSAM

PY - 2006

VL - 100

IS - 1-2

SP - 75

EP - 100

AB - In this paper we discuss the problem of when the projective tensor product of two Banach spaces has the Radon-Nikodym property. We give a detailed exposition of the famous examples of Jean Bourgain and Gilles Pisier showing that there are Banach spaces X and Y such that each has the Radon-Nikodym property but for which their projective tensor product does not; this result depends on the classical theory of absolutely summing, integral and nuclear operators, as well as the famous Grothendieck inequality for its punch-line. In the last section of this paper we discuss many results of a positive character, due to Qingying Bu and various of his coauthors; in particular, we mention results of Bu, Diestel, Dowling and Oja to the effect that if one of the spaces has a boundedly complete FDD then the projective tensor product of two spaces with the RNP has it and a modification of a result of Bu and Pei-Kee Lin to the effect that if X is a Banach lattice with RNP and Y is any Banach space with RNP then their projective product has RNP.

LA - eng

KW - projective tensor product; Radon-Nikodym property; Bourgain-Pisier space; Banach lattice

UR - http://eudml.org/doc/41644

ER -

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