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).

We prove a necessary and sufficient condition for embeddability of an operator system into O 2 . Using Kirchberg's theorems on a tensor product of O 2 and O∞, we establish results on their operator system counterparts S 2 and S∞. Applications of the results proved, including some examples describing C * -envelopes of operator systems, are also discussed.

Abstract. We extend the λ-theory of operator spaces given in [4], that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach * -algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to λ for the algebraic tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of λ-tensor product of C * -algebras has also been discussed.

No abstract

No abstract