Operator space and operator system analogs of Kirchberg's nuclear embedding theorem

Abstract

The Gurarij operator space $\mathbb{NG}$ introduced by Oikhberg is the unique separable $1$-exact operator space that is approximately injective in the category of $1$-exact operator spaces and completely isometric linear maps. We prove that a separable operator space $X$ is nuclear if and only if there exist a linear complete isometry $\varphi :X\rightarrow \mathbb{NG}$ and a completely contractive projection from $\mathbb{NG}$ onto the range of $\varphi $. This can be seen as the operator space analog of Kirchberg's nuclear embedding theorem. We also establish the natural operator system analog of Kirchberg's nuclear embedding theorem involving the Gurarij operator system $\mathbb{GS}$.