The complete separable extension property

TL;DR

This paper develops operator space analogues of the Separable Extension Property, introduces new proofs and results for non-separable cases, and explores structural properties and open problems in this area.

Contribution

It introduces the Complete Separable Extension and Complemention Properties for operator spaces, extending classical results and providing new structural insights and proofs.

Findings

$(igoplus_{n=1}^ fty Z_n)_{c_0}$ has the $(2+\varepsilon)$-SEP for all $\varepsilon>0$ if $Z_n$ have the 1-SEP

$c_0(\ell^\infty)$ has the SEP

$(\bigoplus_{n=1}^\infty Z_n)_{c_0}$ has the CSEP if $Z_n$ are uniformly exact and injective

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.