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

Following Davie's example of a Banach space failing the approximation prope rty (1973), we show how to construct a Banach space E which is asymptotical ly Hilbertian and fails the approximation property. Moreover, the space E i s shown to be a subspace of a space with an unconditional basis which is "a lmost" a weak Hilbert space and which can be written as the direct sum of t wo subspaces all of whose subspaces have the approximation property.