Operator System Nuclearity via -Envelopes

Abstract

We prove that an operator system is (min, ess)-nuclear if its $C^{\ast }$-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] 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 (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).

A non-commutative unitary analogue of Kirchberg's conjecture

“…For the second statement, we will show that U nc (n) satisfies the universal property of C * e (V n ) (see, for example, [13]). Let A be any unital C * -algebra equipped with a unital complete order embedding ι : V n → A such that C * (ι(V n )) = A.…”

Embeddings and -Envelopes of Exact Operator systems

“…(ii) A unital C * -algebra is nuclear if and only if it is exact and has DCEP (see [18,Section 17] and [11,Section 7]) and nuclearity of the C * -envelope implies (min, ess)nuclearity of the operator system [7,Proposition 4.2]. Thus, Theorem 3.1 implies the result.…”

“…Nuclearity properties of operator systems have been characterised in terms of various intrinsic properties (see [9]), and the relation with the nuclearity of their C *envelope was studied in [7]). The double commutant expectation property (DCEP) was introduced in [11] as a generalisation of the weak expectation property (WEP).…”

“…(iii) For an operator system having enough unitaries in C * e (S), (min, ess)-nuclearity is equivalent to nuclearity of C * e (S) [7,Theorem 4.3], so (iii) follows from (ii).…”

Crossed products of operator systems

“…The following proposition is a generalization of a result of Gupta Proof. If C is nuclear, then by [17,Proposition 4.2], S is (min, ess)-nuclear. Conversely, suppose that S is (min, ess)-nuclear, and let D be any unital C * -algebra.…”