The Approximation Property Does Not Imply the Bounded Approximation Property
T. Figiel, W. B. Johnson · Proceedings of the American Mathematical Society · 1973
There is a Banach space which has the approximation property but fails the bounded approximation property. The space can be chosen to have separable conjugate, hence there is a nonnuclear operator on the space which has nuclear adjoint. This latter result solves a problem of Grothendieck [2],