A universal nuclear operator system

Abstract

By means of Fraïssé theory for metric structures developed by Ben Yaacov, we show that there exists a separable $1$-exact operator system $\mathbb{GS}$---which we call the Gurarij operator system---of almost universal disposition. This means that whenever $E\subset F$ are finite-dimensional $1$-exact operator systems, $\varphi :E\to \mathbb{GS}$ is a unital complete isometry, and $\epsilon >0$, there is a linear extension $\hat{\varphi }:F\to \mathbb{GS}$ of $\varphi$ such that $||\hat{\varphi }|{|}_{cb}||{\hat{\varphi }}^{-1}|{|}_{cb}\le 1+\epsilon$. Such an operator system is unique up to complete order isomorphism. Furthermore it is nuclear, homogeneous, and any separable $1$-exact operator system admits a complete order embedding into $\mathbb{GS}$. The space $\mathbb{GS}$ can be regarded as the operator system analog of the Gurarij operator space $\mathbb{NG}$ introduced by Oikhberg, which is in turn a canonical operator space structure on the Gurarij Banach space. We also show that the canonical $\ast$-homomorphism from the universal C*-algebra of $\mathbb{GS}$ to the C*-envelope of $\mathbb{GS}$ is a $\ast$-isomorphism. This implies that $\mathbb{GS}$ does not admit any complete order embedding into a unital exact C*-algebra. In particular $\mathbb{GS}$ is not completely order isomorphic to a unital C*-algebra. With similar methods we show that the Gurarij operator space $\mathbb{NG}$ does not admit any completely isometric embedding into an exact C*-algebra, and in particular $\mathbb{NG}$ is not completely isometric to a C*-algebra. This answers a question of Timur Oikhberg.

This paper has been subsumed by arXiv:1510.05188, and it will not be submitted for publication

References in corpus (1)

• Extremely amenable groups via continuous logic

Cited by in corpus (2)

• Operator space and operator system analogs of Kirchberg's nuclear embedding theorem
• Embeddings and $C^{\ast }$-envelopes of exact operator systems