2018-07-172018-07-17May 20162016-05May 2016Portions of this document have appeared in: Paulsen, Vern I., and Da Zheng. "Tensor products of the operator system generated by the Cuntz isometries." Journal of Operator Theory 76, no. 1 (2016): 67-91.http://hdl.handle.net/10657/3275In 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 ${S}_n^d$.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Cuntz IsometriesOperator SystemsOperator System QuotientsOperator System Tensor ProductsC*-nuclearityComplete Order Isomorphism2018-07-17Thesisborn digital