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

TL;DR

This paper extends Kirchberg's nuclear embedding theorem to operator spaces and systems using the Gurarij operator space and system, providing new characterizations of nuclearity in these contexts.

Contribution

It proves operator space and system analogs of Kirchberg's nuclear embedding theorem using the Gurarij operator space and system.

Findings

Characterization of nuclear operator spaces via the Gurarij operator space.

Operator system analog of Kirchberg's theorem involving Gurarij operator system.

Establishment of a natural operator system analog of the 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 $\mathbb{GS}$.