Tensor Products of the Operator System Generated by the Cuntz Isometries

TL;DR

This paper investigates the tensor product and nuclearity properties of the operator system generated by Cuntz isometries, providing new proofs and characterizations related to Cuntz algebras and operator theory.

Contribution

It establishes the $C^*$-nuclearity of the operator system generated by Cuntz isometries using both nuclearity of the Cuntz algebra and direct methods, offering new insights and proofs.

Findings

$ ext{S}_n$ is $C^*$-nuclear.

Dual operator system of $ ext{S}_n$ embeds into $M_{n+1}$.

A lifting result for Popescu's joint numerical radius.

Abstract

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.