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Article Dans Une Revue Journal of Functional Analysis Année : 2006

Classification of contractively complemented Hilbertian operator spaces

Éric Ricard
Bernard Russo
  • Fonction : Auteur
Matthew Neal
  • Fonction : Auteur

Résumé

We construct some separable infinite dimensional homogeneous Hilbertian operator spaces $H_\infty^{m,R}$ and $H_\infty^{m,L}$, 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 $H_\infty^{m,R}$, $H_\infty^{m,L}$, $H_\infty^{m,R}\cap H_\infty^{m,L}$, or the space $\Phi$ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that $H_{\infty}^{m,L}$ (resp. $H_\infty^{m,R}$) 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 \cite{NeaRus05}, 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 $A^{\ast\ast}$ onto a Hilbertian space which is completely isometric to $R$, $C$, $R \cap C$, or $\Phi$. %Together with \cite{NeaRus05}, this gives a full operator space %classification of 1-complemented Hilbertian operator spaces that This generalizes the well known result, first proved for $B(H)$ by Robertson in \cite{R}, that all Hilbertian operator spaces that are {\it 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 $\Phi$.

Dates et versions

hal-00475262 , version 1 (21-04-2010)

Identifiants

Citer

Éric Ricard, Bernard Russo, Matthew Neal. Classification of contractively complemented Hilbertian operator spaces. Journal of Functional Analysis, 2006, 237 (2), pp.589--616. ⟨10.1016/j.jfa.2006.01.008⟩. ⟨hal-00475262⟩
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