Classification of contractively complemented Hilbertian operator spaces

Abstract

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.