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Journal of Operator Theory

Volume 76, Issue 1, Summer 2016  pp. 67-91.

Tensor products of the operator system generated by the Cuntz isometries

Authors:  Vern I. Paulsen (1) and Da Zheng (2)
Author institution: (1) Department of Pure Mathematics and Institute for Quantum Computing, University of Waterloo, Canada
(2) Department of Mathematics, University of Houston, U.S.A.

Summary:  We study tensor products and nuclearity-related properties of the operator system $\mathcal S_n$ generated by the Cuntz isometries. By using the nuclearity of the Cuntz algebra, we can show that $\mathcal{S}_n$ is $C^*$-nuclear, and this implies a dual row contraction version of Ando's theorem characterizing operators of numerical radius 1. On the other hand, without using the nuclearity of the Cuntz algebra, we are still able to show directly this Ando type property of dual row contractions and conclude that $\mathcal{S}_n$ is $C^*$-nuclear, which yields a new proof of the nuclearity of the Cuntz algebras. We prove that the dual operator system of $\mathcal{S}_n$ is completely order isomorphic to an operator subsystem of $M_{n+1}$. Finally, a lifting result concerning Popescu's joint numerical radius is proved via operator system techniques.

Keywords:  Cuntz isometries, operator system tensor product, $C^*$-nuclearity, operator system quotient, dual row contraction, shorted operator, joint numerical radius

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