Operator system quotients of matrix algebras and their tensor products

Abstract

If φ : S → T is a completely positive (cp) linear map of operator systems and if J = ker φ, then the quotient vector space S/J may be endowed with a matricial ordering through which S/J has the structure of an operator system. Furthermore, there is a uniquely determined cp maṗ φ : S/J → T such that φ =φ • q, where q is the canonical linear map of S onto S/J . The cp map φ is called a complete quotient map ifφ is a complete order isomorphism between the operator systems S/J and T . Herein we study certain quotient maps in the cases where S is a full matrix algebra or a full subsystem of tridiagonal matrices.Our study of operator system quotients of matrix algebras and tensor products has applications to operator algebra theory. In particular, we give a new, simple proof of Kirchberg's Theoremshow that an affirmative solution to the Connes Embedding Problem is implied by various matrix-theoretic problems, and give a new characterisation of unital C * -algebras that have the weak expectation property.

A non-commutative unitary analogue of Kirchberg's conjecture

“…To show that V n is a complete quotient of M 2n , we need some equivalent characterizations of positivity in the quotient operator system M 2n /J 2n . This result and its proof are analogous to [11,Proposition 2.3].…”

“…Using the notation of [11], whenever i, j ∈ {1, ..., 2n} are such that ϕ(E ij ) = 0, we define e ij =Ė ij ∈ M 2n /J 2n . We let q : M 2n → M 2n /J 2n be the canonical quotient map.…”

“…Proof. We use the same argument as in [11,Proposition 2.7] 2n to be the canonical normalized trace, this mapping is a unital complete order isomorphism. It follows that the vector space dual of M 2n /J 2n , equipped with the operator system structure inherited from M 2n , is the operator system dual of V n .…”

Crossed products of operator systems

“…Clearly, the flip map S ⊗ T → T ⊗ S induces a complete order isomorphism S ⊗ ue T → T ⊗ eu S. On the other hand, we can at least distinguish ue from c. To this end, we need a slight generalization of a result of Kavruk [26,Corollary 5.8]. The proof is almost identical to [15,Proposition 3.6]; we include it for completeness. Proposition 4.8.…”

Lovász theta type norms and operator systems

“…If we let S = M n and let J denote the set of diagonal matrices of trace 0, then J is a kernel and it follows from the characterization of the quotient M n /J in [4] that n ≤ δ(M n , J ). For any J ∈ J we see that tr(I n + J) = n and so when I n + J ≥ 0 we see that I n + J ≤ n. Letting J be the diagonal matrix with diagonal entries, (n − 1, −1, .…”