, Volume 23, Issue 9, pp 1697-1706
Date: 14 Feb 2007

Some Properties of the Injective Tensor Product of Banach Spaces

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Abstract

Let X and Y be Banach spaces such that X has an unconditional basis. Then X \( \overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{ \otimes } \) Y , the injective tensor product of X and Y , has the Radon–Nikodym property (respectively, the analytic Radon–Nikodym property, the near Radon–Nikodym property, non-containment of a copy of c 0, weakly sequential completeness) if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.

The first author is supported by the National Natural Science Foundation of China, Grant No. 10571035
The third author gratefully acknowledges the support of the International Office of Harbin Institute of Technology for his visit to the Department of Mathematics, Harbin Institute of Technology from mid-May to mid-June in 2005 and the support from the Summer Research Grant of the College of Liberal Arts, the University of Mississippi in the summer of 2005