UC Irvine

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 Hnk, 1 ≤ k ≤ n, generalizing the row and column Hilbert spaces Rn and Cn, and we show that an atomic subspace X ⊂ B(H) that is the range of a contractive projection on B(H) is isometrically completely contractive to an l∞-sum of the Hnk 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 ideal.