The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja\u27s conjecture. As a consequence, we show that each of the spaces c0 and ℓ1 has a subspace which has the AP but fails to have the strong AP. © 2013 Elsevier Inc