Abstract
We define a family of Hilbertian operator spaces H n k , 1≤ k≤ n, containing the row and column Hilbert spaces R n , C n and show that an atomic subspace X⊂B( H) which is the range of a contractive projection on B( H) is isometrically completely contractive to an ℓ ∞-sum of the H n k and Cartan factors of types 1 to 4. We also give a classification up to complete isometry of w ∗ -closed atomic JW ∗ -triples which have no infinite-dimensional rank 1 w ∗ -closed ideal.
Highlights
It was shown by Choi-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 which are atomic
Cartan factors of types 5 and 6 will play no role in this paper since neither is even isometric to a 1mixed injective operator space
Summary
It was shown by Choi-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-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]. 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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