An example of an asymptotically Hilbertian space which fails the approximation property

Following Davie's example of a Banach space failing the approximation property [D], we show how to construct a Banach space E which is asymptotically Hilbertian and fails the approximation property. Moreover, the space E is shown to be a subspace of a space with an unconditional basis which is ``almost'' a weak Hilbert space and which can be written as the direct sum of two subspaces all of whose subspaces have the approximation property.