# Journal of Operator Theory

Volume 43, Issue 2, Spring 2000 pp. 329-374.

The complete separable extension property**Authors**: Haskell Rosenthal

**Author institution:**Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA

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

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