Operator System Nuclearity via $C^*$-envelopes

TL;DR

This paper establishes a connection between the nuclearity of operator systems and their C*-envelopes, providing characterizations related to group amenability and graph completeness.

Contribution

It proves that (min, ess)-nuclearity of an operator system is equivalent to the nuclearity of its C*-envelope and characterizes such nuclearity for group and graph operator systems.

Findings

Operator system (min, ess)-nuclearity iff C*-envelope is nuclear.

Group operator system (min, ess)-nuclear iff the group is amenable.

Operator system (min, max)-nuclearity for groups of order ≤ 3 and complete graphs.

Abstract

We prove that an operator system is (min, ess)-nuclear if its C*-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of countable discrete group by Farenick et al. is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (resp., a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to 3 (resp., every component of the graph is complete).