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 H_n^k , 1 ≤ k ≤ n 1\le k\le n , generalizing the row and column Hilbert spaces R n R_n and C n C_n , and we show that an atomic subspace X ⊂ B ( H ) X\subset B(H) that is the range of a contractive projection on B ( H ) B(H) is isometrically completely contractive to an ℓ ∞ \ell ^\infty -sum of the H n k H_n^k and Cartan factors of types 1 to 4. In particular, for finite-dimensional X 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 J W ∗ JW^* -triples without an infinite-dimensional rank 1 w ∗ ^* -closed ideal.
Highlights
It was shown by Choi and Effros that an injective operator system is isometric to a conditionally complete C∗-algebra [6, Theorem 3.1]
We provide in Theorem 2 a classification up to isometric complete contraction of 1-mixed injectives that are atomic
Cartan factors of types 5 and 6 will play no role in this paper, since neither is even isometric to a 1-mixed injective operator space
Summary
It was shown by Choi and Effros that an injective operator system is isometric to a conditionally complete C∗-algebra [6, Theorem 3.1]. A special case of a result of Friedman and Russo showed that if a projection on a C∗-algebra is contractive, the range is isometric to a Banach Jordan triple system [13, Theorem 2]. Robertson [30, Corollary 3] proved that an injective operator space that is isometric to 2 is completely isometric to R or C, where R and C denote the row and column operator space versions of 2 These results can be thought of as giving a partial classification of injectives up to various types of isomorphisms. We show that an atomic (in particular, finite-dimensional) contractively complemented subspace of a C∗-algebra is a 1-mixed injective, that is, the range of a contractive projection on some B(H) Most of these results have been announced in [25]
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