Classification of contractively complemented Hilbertian operator spaces
Published Web Locationhttps://doi.org/10.1016/j.jfa.2006.01.008
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 finite-dimensional 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.