A solution of one Problem of Lindenstrauss and Rosenthal on Subspace   Homogeneous and Quotient Homogeneous Banach Spaces with Application to the   Approximation Problem and to the Schroeder - Bernstein Problem
  In article is constructed a wide couple of pairwice non-isomorphic separable superreflexive Banach spaces E that are subspace homogeneous. Their conjugates are quotient homogeneous. None of this couple neither isomorphic to its Cartesian square nor has the approximation property. At the same time, any such E is isomorphic to E+ E+ W for some Banach space W and, hence, solves the Schroeder - Bernstein problem. 
Eugene Tokarev