On certain extension properties for the space of compact operators
| dc.creator | Oikhberg, Timur | |
| dc.creator | Rosenthal, Haskell P. | |
| dc.date | 1999-05-04 | |
| dc.date.accessioned | 2026-07-07T05:28:56Z | |
| dc.date.available | 2026-07-07T05:28:56Z | |
| dc.description | 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. | |
| dc.description | 71 pages, AMSTeX | |
| dc.identifier | https://arxiv.org/abs/math/9905017 | |
| dc.identifier | http://arxiv.org/abs/math/9905017 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78452 | |
| dc.subject | Operator Algebras | |
| dc.subject | 46B03, 46B28, 47D25 (Primary) 47C15, 46L99 (Secondary) | |
| dc.title | On certain extension properties for the space of compact operators | |
| dc.type | text |