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Operator spaces with few completely bounded maps

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Abstract

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 X 0 such that, whenever X’ is an infinite dimensional quotient of X 0 , X is a subspace of X’, and \({{T : X {{\rightarrow}} X'}}\) 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 X 0 fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA

    Timur Oikhberg

  2. Département de Mathématiques de Besançon, Université de Franche-Comté, 25030, Besançon cedex, France

    Éric Ricard

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  1. Timur Oikhberg
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  2. Éric Ricard
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Correspondence to Timur Oikhberg.

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Mathematics Subject Classification (2000):

The first author was supported in part by the NSF grants DMS-9970369, 0296094, and 0200714.

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Oikhberg, T., Ricard, É. Operator spaces with few completely bounded maps. Math. Ann. 328, 229–259 (2004). https://doi.org/10.1007/s00208-003-0481-2

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  • DOI: https://doi.org/10.1007/s00208-003-0481-2

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