ar X iv : m at h . O A / 9 80 40 64 v 1 14 A pr 1 99 8 THE COMPLETE SEPARABLE EXTENSION PROPERTY

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 + ε)-SEP for all ε > 0 if Z1, Z2, . . . have the 1-SEP; in particular, c0(l∞) has the SEP. It is proved that e.g., c0(R⊕C) has the CSEP (where R, C 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., K0 def = (⊕∞ 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.