1998/04/14 by Haskell P. Rosenthal, Rosenthal, Haskell P. · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Holomorphic and Operator Theory #math.FA #math.OA #msc:46B15 #msc:47D25
paper · pdf · doi:10.48550/arxiv.math/9804064
56 pages
arxiv created 1998/04/14 · arxiv updated 2009/11/30
This work introduces operator space analogues of the Separable Extension Property (SEP) for Banach spaces; the Complete Separable Extension Property (CSEP) and the Complete Separable Complemention Property (CSCP). The results use the technique of a new proof of Sobczyk's Theorem, which also yields new results for the SEP in the non-separable situation, e.g., (⊕n=1^∞ Zn)c0 has the (2+\ep)-SEP for all \ep>0 if Z1,Z2,... have the 1-SEP; in particular, c0 (ℓ^∞) has the SEP. It is proved that e.g., c0(\bR⊕\bC) has the CSEP (where \bR, \bC denote Row, Column space respectively) as a consequence of the general principle: if Z1,Z2,... is a uniformly exact sequence of injective operator spaces, then (⊕n=1^∞ Zn)c0 has the CSEP. Similarly, e.g., \bK0 \defeq (⊕n=1^∞ Mn)c0 has the CSCP, due to the general principle: (⊕n=1^∞ Zn)c0 has the CSCP if Z1,Z2,... are injective separable operator spaces. Further structural results are obtained for these properties, and several open problems and conjectures are discussed.