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The projective tensor product. II: The Radon–Nikodym property. (English) Zbl 1118.46030

In 1983, J.Bourgain and G.Pisier constructed a Banach space \(X\) with the RNP, indeed an \({\mathcal L}_\infty\)-space, such that the projective tensor product \(X \widehat{\otimes}_\pi X\) contains a copy of \(c_0\) and hence fails the RNP. This result was published in [Bol.Soc.Bras.Mat.14, No.2, 109–123 (1983; Zbl 0586.46011)], a source which is not easily accessible.
In the present paper, the authors give a faithful and detailed exposition of this result, and they also mention positive results by Q.Bu, N.J.Kalton, E.Oja and others giving conditions when the RNP does pass from \(X\) and \(Y\) to \(X \widehat{\otimes}_\pi Y\).
The reader should be cautioned that the authors’ definition of an \(\eta\)-admissible embedding is too sloppy; Bourgain and Pisier are much more precise at this point. (The definition should refer to the universal property exhibited on p.79 rather than the mere statement of Theorem 1.) Also, the authors display two lemmas from Q.Bu’s and P.–K.Lin’s paper [J. Math.Anal.Appl.293, No.1, 149–159 (2004; Zbl 1054.46015)], but they mention the main result of this paper only casually between the lines. Unfortunately, the number of typos is not negligible.
[For Part I by the same authors, see A.Kaminska (ed.), Trends in Banach spaces and operator theory. Contemp.Math.321, 37–65 (2003; Zbl 1048.46024).]

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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