1999/05/04 by Timur Oikhberg, Oikhberg, Timur, Haskell P. Rosenthal +1
Mathematics · #46B03 #46B28 #46L99 (Secondary) #47D25 (Primary) 47C15 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46B03 #msc:46B28 #msc:46L99 #msc:47C15 #msc:47D25
paper · pdf · doi:10.48550/arxiv.math/9905017
71 pages, AMSTeX
arxiv created 1999/05/04 · arxiv updated 2009/11/30
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; 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 K0 the c0 sum of \Cal Mn'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 K0 has this property. A new proof is given for the result (due to E. Kirchberg) that \bf K0 (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 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.