OPERATOR SYSTEM QUOTIENTS OF MATRIX ALGEBRAS AND THEIR TENSOR PRODUCTS

If phi : S -> T is a completely positive (cp) linear map of operator systems and if J = ker phi, 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 map (phi) over dot : S/J -> T such that phi = (phi) over dot o q, where q is the canonical linear map of S onto S/J. The cp map phi is called a complete quotient map if (phi) over dot 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 Theorem C*(F-infinity) circle times(min) B(H) = C*(F-infinity) circle times(max) B(H), show 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.