The Operator System Generated by Cuntz Isometries and its Applications

In this thesis, we focused on the operator system, $Sn$, generated by $n$ () Cuntz isometries, i.e. $Sn=span\{I,Si,Si\ast :1\le i\le n\}$. We first studied the properties of $Sn$, such as the uniqueness, the universal property and the embedding property. Then we constructed an operator subsystem $En$ in $Mn$---the $n$ by $n$ matrix algebra and proved that $Sn$ is completely order isomorphic to an operator system quotient of $En$. This result also led to a characterization of positive elements in $Sn$.

Next, we studied the tensor products and related properties of $Sn$, which was motivated by the nuclearity of the Cuntz algebra $On$. In contrast with $On$, $Sn$ is not nuclear in the operator system category. However, we could show that it is $C\ast$-nuclear by using the nuclearity of $On$ and some dilation theoerems. This implied an Ando-type theorem for dual row contractions. With the help of shorted operator techniques, we were able to show that $Sn$ is $C\ast$-nuclear without using the nuclearity of $On$. And this provided us with a new proof of the nuclearity of the $On$.

Finally, we turned our attention to the dual operator system $Snd$ of $Sn$. By considering $Snd$, we were able to derive an alternative characterization of the dual row contractions as well as an equivalent condition for unital completely positive maps on $Snd$. Moreover, it was a little surprising to see that $Snd$ is completely order isomorphic to $En\prime$, an operator subsystem in $Mn+1$. The last result was a lifting theorem about the joint numerical radius, which was implied by the $C\ast$-nuclearity of $Snd$.