Operator spaces with few completely bounded maps

Timur Oikhberg, Éric Ricard

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)229-259
Number of pages31
JournalMathematische Annalen
Issue number1-2
StatePublished - Jan 2004
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)


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