The weakly compact approximation of the projective tensor product of Banach spaces

Abstract

We show that X ^ ›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 ^ ›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) ⁄^ ›Y has the W.A.P.