Classification of Contractively Complemented Hilbertian Operator Spaces

We construct some separable infinite dimensional homogeneous Hilbertian operator spaces H ∞ and H m,L ∞ , which generalize the row and column spaces R and C (the case m = 0). We show that separable infinitedimensional Hilbertian JC∗-triples are completely isometric to an element of the set of (infinite) intersections of these spaces . This set includes the operator spaces R, C, R ∩ C, and the space Φ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that H ∞ (resp. H m,L ∞ ) is completely isometric to the space of creation (resp. annihilation) operators on the m (resp. m + 1) anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in [14], this gives a full operator space classification of all rank-one JC∗-triples in terms of creation and annihilation operator spaces. We use the above to show that all contractive projections on a C*-algebra A with infinite dimensional Hilbertian range are “expansions” (which we define precisely) of normal contractive projections from A∗∗ onto a Hilbertian space which is completely isometric to R, C, R ∩ C, or Φ. This generalizes the well known result, first proved for B(H) by Robertson in [17], that all Hilbertian operator spaces that are completely contractively complemented in a C*-algebra are completely isometric to R or C. We also compute various completely bounded Banach-Mazur distances between these spaces, or Φ.