Extension Properties for the Space of Compact Operators

✍ Scribed by Timur Oikhberg; Haskell P Rosenthal

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 377 KB
• Volume: 179
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3674

✦ Synopsis

Let Z be a fixed separable operator space, X/Y general separable operator spaces, and T : X Ä Z a completely bounded map. Z is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely bounded extension to Y and the Mixed Separable Extension Property (MSEP) if every such T admits a bounded extension to Y. Finally, Z is said to have the Complete Separable Complementation Property (CSCP) if Z is locally reflexive and T admits a completely bounded extension to Y provided Y is locally reflexive and T is a complete surjective isomorphism. Let K denote the space of compact operators on separable Hilbert space and K 0 the c 0 sum of M n 's (the space of ``small compact operators''). It is proved that K has the CSCP, using the second author's previous result that K 0 has this property. A new proof is given for the result (due to E. Kirchberg) that K 0 (and hence K) fails the CSEP. It remains an open question if K has the MSEP; it is proved this is equivalent to whether K 0 has this property. A new Banach space concept, Extendable Local Reflexivity (ELR), is introduced to study this problem. Further complements and open problems are discussed.

Academic Press

Contents

Introduction.

1. Extending complete isomorphisms into B(H).

2. An operator space construction on certain subspaces of M . 3. The *-mixed separable extension property and extendably locally reflexive banach spaces. 4. K 0 fails the CSEP: a new proof and generalizations.