Parallel to the study of finite dimensional Banach spaces, there is a growing
interest in the corresponding local theory of operator spaces. We define a
family of Hilbertian operator spaces H_n^k,0< k < n+1, generalizing the row and
column Hilbert spaces R_n,C_n and show that an atomic subspace X of B(H) which
is the range of a contractive projection on B(H) is isometrically completely
contractive to a direct sum of the H_n^k and Cartan factors of types 1 to 4. In
particular, for finite dimensional X, this answers a question posed by Oikhberg
and Rosenthal. Explicit in the proof is a classification up to complete
isometry of atomic w*-closed JW*-triples without an infinite dimensional rank 1
w^*-closed idealComment: 40 pages, latex, the paper was submitted in October of 2000 and an
announcement with the same title appeared in C. R. Acad. Sci. Paris 331
(2000), 873-87
