Contractive projections and operator spaces

We define a family of Hilbertian operator spaces H-n(k), 1 less than or equ al to k less than or equal to n, containing the row and column Hilbert spac es R-n, C-n and show that an atomic subspace X subset of B(H) which is the range of a contractive projection on B(H) is isometrically completely contr active to an l(infinity)-sum of the H-n(k) and Cartan factors of types 1 to 4. We also give a classification up to complete isometry of w*-closed atom ic JW*-triples which have no infinite-dimensional rank 1 w*-closed ideal. ( C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.