  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
${\bf K}$ denote the space of compact operators on separable Hilbert space and
${\bf K}_0$ the $c_0$ sum of ${\Cal M}_n$'s (the space of ``small compact
operators''). It is proved that ${\bf K}$ has the CSCP, using the second
author's previous result that ${\bf K}_0$ has this property. A new proof is
given for the result (due to E. Kirchberg) that ${\bf K}_0$ (and hence ${\bf
K}$) fails the CSEP. It remains an open question if ${\bf K}$ has the MSEP; it
is proved this is equivalent to whether ${\bf K}_0$ 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.
