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. 
P. G. Casazza, C. L. Garcia, W. B. Johnson