Extension Properties for the Space of Compact Operators

Let Z be a fixed separable operator space, X⊂Y general separable operator spaces, and T: X→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 K0 the c0 sum of Mnʹs (the space of “small compact operators”). It is proved that K has the CSCP, using the second authorʹs previous result that K0 has this property. A new proof is given for the result (due to E. Kirchberg) that K0 (and hence K) fails the CSEP. It remains an open question if K has the MSEP; it is proved this is equivalent to whether K0 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.