Annales Academić Scientiarum Fennicć
Mathematica
Volumen 37, 2012, 461-465

THE WEAKLY COMPACT APPROXIMATION OF THE PROJECTIVE TENSOR PRODUCT OF BANACH SPACES

Qingying Bu

University of Mississippi, Department of Mathematics
University, MS 38677, U.S.A.; qbu 'at' olemiss.edu

Abstract. We show that X \hat\otimes Y, the projective tensor product of Banach spaces X and Y, has the (bounded) compact approximation property if and only if both X and Y have the same property. We also show that X \hat\otimes Y has the weakly compact approximation property (W.A.P.) if both X and Y has the W.A.P. and either (i) every bounded linear operator from X (resp. from Y) to Y^* (resp. to X^*) is completely continuous, or (ii) one of X and Y has the Dunford-Pettis property. As a consequence, we show that if K is scattered and Y has the W.A.P., then C(K)^* \hat\otimes Y has the W.A.P.

2010 Mathematics Subject Classification: Primary 46B28, 46B42.

Key words: (Weakly) compact approximation property, projective tensor product.

Reference to this article: Q. Bu: The weakly compact approximation of the projective tensor product of Banach spaces. Ann. Acad. Sci. Fenn. Math. 37 (2012), 461-465.

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doi:10.5186/aasfm.2012.3737