On certain extension properties for the space of compact operators

Abstract

Let $Z$ be a fixed separable operator space, $X\subset Y$ general separable operator spaces, and $T:X\to Z$ 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 $Y$; 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 $\mathbf{K}$ denote the space of compact operators on separable Hilbert space and ${\mathbf{K}}_0$ the $c_0$ sum of ${\Cal M}_n$'s (the space of ``small compact operators''). It is proved that $\mathbf{K}$ has the CSCP, using the second author's previous result that ${\mathbf{K}}_0$ has this property. A new proof is given for the result (due to E. Kirchberg) that ${\mathbf{K}}_0$ (and hence $\mathbf{K}$) fails the CSEP. It remains an open question if $\mathbf{K}$ has the MSEP; it is proved this is equivalent to whether ${\mathbf{K}}_0$ 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.

71 pages, AMSTeX