Embeddings and $C^*$-envelopes of exact operator systems

TL;DR

This paper characterizes when an operator system can be embedded into the Cuntz algebra , and explores their $C^*$-envelopes using tensor product theorems, with applications to specific operator systems.

Contribution

It provides a necessary and sufficient condition for embedding operator systems into  and extends Kirchberg's tensor product results to operator system contexts.

Findings

Characterization of embeddability into 

Results on operator system tensor products involving  and 

Examples of $C^*$-envelopes of operator systems

Abstract

We prove a necessary and sufficient condition for embeddability of an operator system into $\mathcal{O}_2$. Using Kirchberg's theorems on a tensor product of $\mathcal{O}_2$ and $\mathcal{O}_{\infty}$, we establish results on their operator system counterparts $\mathcal{S}_2$ and $\mathcal{S}_{\infty}$. Applications of the results proved, including some examples describing $C^*$-envelopes of operator systems, are also discussed.