Classification of contractively complemented Hilbertian operator spaces

We construct some separable infinite-dimensional homogeneous Hilbertian operator spaces H m,R ∞ and H m,L ∞ , which generalize the row and column spaces R and C (the case m = 0). We show that a separable infinite-dimensional Hilbertian JC∗-triple is completely isometric to one of H m,R ∞ , H m,L ∞ , H m,R ∞ ∩ H m,L ∞ , or the space Φ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that H m,L ∞ (respectively H m,R ∞ ) is completely isometric to the space of creation (respectively annihilation) operators on the m (respectively m+1) anti-symmetric tensors of the Hilbert space. Together with the finitedimensional case studied in [M. Neal, B. Russo, Representation of contractively complemented Hilbertian operator spaces on the Fock space, Proc. Amer. Math. Soc. 134 (2006) 475–485], 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 structural result for Hilbertian JC∗-triples 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 [A.G. Robertson, Injective matricial Hilbert spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991) 183–190], that all Hilbertian operator spaces that are completely contractively complemented in a C∗-algebra are completely isometric to R or C. We use the above representation on the Fock space to compute various completely bounded Banach–Mazur distances between these spaces, or Φ. © 2006 Elsevier Inc. All rights reserved.