# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2015aug04.2093

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

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