Contractive projections and operator spaces

Neal, Matthew; Russo, Bernard

doi:10.1090/S0002-9947-03-03233-1

Contractive projections and operator spaces
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Trans. Amer. Math. Soc. 355 (2003), 2223-2262 Request permission

Abstract:

Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces $H_n^k$, $1\le k\le n$, generalizing the row and column Hilbert spaces $R_n$ and $C_n$, and we show that an atomic subspace $X\subset B(H)$ that is the range of a contractive projection on $B(H)$ is isometrically completely contractive to an $\ell ^\infty$-sum of the $H_n^k$ and Cartan factors of types 1 to 4. In particular, for finite-dimensional $X$, this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w$^*$-closed $JW^*$-triples without an infinite-dimensional rank 1 w$^*$-closed ideal.

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