Classification of contractively complemented Hilbertian operator spaces

Abstract

We construct some separable infinite dimensional homogeneous Hilbertian operator spaces which generalize the row and column spaces R and C. We show that separable infinite-dimensional 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, their intersection, and the space spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that these new spaces are completely isometric to the space of creation (resp. annihilation) operators on the anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in our previous paper, 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 the second dual of A onto a Hilbertian space which is completely isometric to one of the four spaces mentioned above. This generalizes the well known result, first proved for B(H) by Robertson, 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.