Browsing by Author "Zheng, Da 1987-"
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Item The Operator System Generated by Cuntz Isometries and its Applications(2016-05) Zheng, Da 1987-; Paulsen, Vern I.; Blecher, David P.; Tomforde, Mark; Smith, RogerIn this thesis, we focused on the operator system, ${S}_n$, generated by $n$ ($2\leq n< \infty$) Cuntz isometries, i.e. ${S}_n=span\{I,S_i,S_i^*:1\leq i\leq n\}$. We first studied the properties of ${S}_n$, such as the uniqueness, the universal property and the embedding property. Then we constructed an operator subsystem ${E}_n$ in $M_n$---the $n$ by $n$ matrix algebra and proved that ${S}_n$ is completely order isomorphic to an operator system quotient of ${E}_n$. This result also led to a characterization of positive elements in ${S}_n$. Next, we studied the tensor products and related properties of ${S}_n$, which was motivated by the nuclearity of the Cuntz algebra ${O}_n$. In contrast with ${O}_n$, ${S}_n$ is not nuclear in the operator system category. However, we could show that it is $C^*$-nuclear by using the nuclearity of ${O}_n$ 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 ${S}_n$ is $C^*$-nuclear without using the nuclearity of ${O}_n$. And this provided us with a new proof of the nuclearity of the ${O}_n$. Finally, we turned our attention to the dual operator system ${S}_n^d$ of ${S}_n$. By considering ${S}_n^d$, 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 ${S}_n^d$. Moreover, it was a little surprising to see that ${S}_n^d$ is completely order isomorphic to ${E}'_n$, an operator subsystem in $M_{n+1}$. The last result was a lifting theorem about the joint numerical radius, which was implied by the $C^*$-nuclearity of ${S}_n^d$.