The complete separable extension property

Abstract

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., $(\oplus_{n=1}^\infty Z_n)_{c_0}$ has the $(2+\ep)$-SEP for all $\ep>0$ if $Z_1,Z_2,...$ have the 1-SEP; in particular, $c_0 (\ell^\infty)$ has the SEP. It is proved that e.g., $c_0(\bR\oplus\bC)$ has the CSEP (where $\bR$, $\bC$ denote Row, Column space respectively) as a consequence of the general principle: if $Z_1,Z_2,...$ is a uniformly exact sequence of injective operator spaces, then $(\oplus_{n=1}^\infty Z_n)_{c_0}$ has the CSEP. Similarly, e.g., $\bK_0 \defeq (\oplus_{n=1}^\infty M_n)_{c_0}$ has the CSCP, due to the general principle: $(\oplus_{n=1}^\infty Z_n)_{c_0}$ has the CSCP if $Z_1,Z_2,...$ are injective separable operator spaces. Further structural results are obtained for these properties, and several open problems and conjectures are discussed.