We construct several examples of Hilbertian operator spaces with few completely bounded maps. In particular, we give an example of a separable 1-Hilbertian operator space X0 such that, whenever X′ is an infinite dimensional quotient of X0, X is a subspace of X′, and T : X→X′ is a completely bounded map, then T = λI X + S, where S is compact Hilbert-Schmidt and ||S||2/16 ≤ ||S||cb ≤ ||S||2. Moreover, every infinite dimensional quotient of a subspace of X0fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property.
|Original language||English (US)|
|Number of pages||31|
|State||Published - Jan 2004|
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